1000 Sentences With "topology" | Random Sentence Generator

Andret studies a branch of mathematics called topology, and the book's architecture mimics . . . topology.

The company's new so-called Pegasus topology connects every qubit to 15 other qubits, up from six in its current topology.

I used to do very pure mathematics — symplectic topology.

Actually, I bet a gamer could speed-run that four-dimensional topology.

K-theory studies all aspects of that situation—the topology and the geometry.

Thors Hans Hansson: Topology is a branch of mathematics—a pretty abstract branch.

The area is also landlocked and surrounding topology can act as a basin.

Topological quantum computing exploits the field of geometry known as topology, hence its name.

They could make out a volcano, then a port — the topology of Central America.

His 1983 theorem, the Johnson homomorphism, is still studied in the field of geometric topology.

And unless you're on that very tall spire, it's not obvious what the topology is.

Tonight at AWS re:Invent, the company announced a new tool called AWS Transit Gateway designed to help build a network topology inside of AWS that lets you share resources across accounts and bring together on premises and cloud resources in a single network topology.

The idea is to help find people the products that are best for their skin topology.

" Topology just launched in July and says it has already had orders in the "low hundreds.

The topology of threshold — every breakthrough another base and divide, limit and ceiling, line and maximum.

The topology of a Dinara Kasko cake is difficult to explain without discursions into higher math.

New Foundations rethinks topology in a way that allows for a clearer distinction between time and space.

Noether continued doing vital mathematical work in abstract algebra and topology all through the 1920s and 1930s.

Whether the algorithms of computational topology would show any similarities between those locales is an intriguing question.

They rarely follow any strict convention and often result in changes to lane markings or road topology.

As previously mentioned, the dark web is a shifting topology, and things can be difficult to find.

Forty years ago, physics and the study of geometry and topology had little to do with one another.

Topology Eyewear is an augmented reality app providing custom-fit glasses from a 3D scan of your face.

Each pair of Topology glasses start at $495 and go upward to $800, depending on the add-ons.

The party hasn't won statewide office since 21966, and gerrymandered congressional districts are a realpolitik of entangled topology.

It also depends to some extent on the shape of the material - the topology, as we say in physics.

The moment of birth, the moment of death, a topology of the universe, it all depends on the interval.

Topology is a branch of math that studies what properties are preserved when objects are stretched, twisted, or deformed.

But when I went to college at Harvard, I took a course in topology, which is the study of spaces.

The key to their success was something called topology, a branch of mathematics focused on the fundamental shapes of things.

Other variables, like a city's topology and enforcement of drunk driving laws, can also effect whether residents use ride-hailing services.

And he finds that, to match the results of standard topology, the lines need to be directed, just as time is.

You can check out how Topology works in the video below or download and test its tech yourself on the app.

AWS Transit Gateway lets you build connections across a network wherever the resources live in a standard kind of network topology.

Picardi browsed developments on Google Earth to grasp their geography and topology, and strolled amidst them virtually on Google Street View.

Theoreticians wanted to explain Dr. Klitzing's discovery mathematically, so Dr. Thouless, working with three research assistants, applied topology to the problem.

If you really want to recreate the surface of the Moon you need information on the topology of its surface as well.

Their research centers on topology, a branch of mathematics involving step-wise changes like making a series of holes in an object.

The new study from a team of researchers in the United States and the United Kingdom took a look at the Martian topology.

The main difference between what Microsoft is doing is that its system is based on advances in topology that the company previously discussed.

Conventionally, topology—the first level of geometrical structure—is defined using open sets, which describe the neighborhood of a point in space or time.

To avoid this, Dr Silva turned to computational topology—a field that finds algorithms to describe complicated shapes and surfaces as simply as possible.

Because I couldn't clone my brain, I found myself sitting, late one afternoon, in "Vibrations, Scale, and Topology," where a musician from Tulsa, Okla.

GOAT stands for Gearless Omni-directional Acceleration-vectoring Topology and uses a single motor and planetary gear to create a highly dynamic range of motion.

Dr. Atiyah's early work was in topology, a field of mathematics that studies shape, including that of mathematical objects with many more than three dimensions.

Rather than shipping myriad glasses your way, Topology lets you see what you'd look like in a variety of frame styles and colors using augmented reality.

"What topology does is that it gives you this ability to have much better fidelity," Microsoft's corporate vice president for quantum research Todd Holmdahl told me.

Hansson, apparently anticipating our total ignorance of topology, helpfully brought along a cinnamon bun, a bagel, and a pretzel to explain it at the prize announcement.

Computational topology is already employed in tasks as diverse as loading goods at dockyards and studying the way protein molecules fold, so many topological algorithms already exist.

But there are certain aspects of topology which are pretty easy to understand, at least the basic concepts like the topological invariant and its applications in physics.

New research published Monday in Nature Geoscience suggests it was likely a stationary gravity wave generated by the flow of air through the planet's mountainous surface topology.

Peter Lin, sales director of Topology Travel, an agency in Taipei, said that under the policy, inquiries from the Philippines, Thailand and Vietnam had more than doubled.

Although they couldn't find the exact solutions to any differential equation, they did manage to use topology to reveal the number of solutions such an equation has.

I was trying to explain standard topology to some students when I was teaching a class on space and time, and I realized that I didn't understand it.

One option would be to launch rockets from the highest mountain tops, but given the flat topology of super-Earths, tall mountains would be hard to come by.

That is the essence of topology, a branch of geometry which deals in "invariants", such as holes, that can exist in geometric shapes only in discrete, integer numbers.

But unlike the states, one of the biggest barriers comes down to topology: they just literally can't get the wires to reach across the rocky hills and valleys.

Zheng measures society's power topology through its negative spaces: where the rules are not; where games can become reality, even just for the sake of the game itself.

He is working on a computer simulation of knitted fabric, inputting yarn properties and stitch topology, and outputting the geometry and elasticity of the real-life finished object.

And needless to say, he found that for the three-body problem, this was very complicated—though in his efforts to analyze it he invented the field of topology.

It's the general form of the shape—the topology—that's important, so researchers are keen to find the designs that use the smallest number of qubits while maximizing performance.

While they constitute a small body of work within Johns's diverse oeuvre, they possess a topology that shares something with other bodies of work across a variety of mediums.

"It's a metaphor for how we're uploading our lives into all these corporate and government databases… you're literally digitizing the topography and topology of their body," says the artist.

"Because of the network topology and structure, most issues can be resolved remotely and in the instance that there is an issue, we often have significant redundancy," he said.

Though such objects can't be visualized, topology provides tools to figure out how many holes they have and how different parts of an object are connected to one another.

The researchers of the Facebook-MIT system believe their system is superior because, rather than addresses that are arbitrary and meaningless, their assigned addresses adhere to current roads and topology.

When Mr. Canin talks about Milo's study of topology, a mathematical field that studies geometric properties and spatial relations between flexible objects, he could easily be describing his own work.

Maudlin's approach differs from other approaches that extend standard topology to endow geometry with directionality; it is not an extension, but a rethinking that builds in directionality at the ground level.

We've seen photos and maps of Pluto and Charon before, but these maps now represent our best current understanding of the surface features and topology of these distant Kuiper Belt objects.

A graduate of the Massachusetts Institute of Technology who suspended his graduate studies in algebraic topology to launch BEAM, Mr. Zaharopol had some idea of what his students were up against.

Subfields of topology include geometric topology, differential topology, algebraic topology and general topology.

Introduction to topology: pure and applied. Pearson Prentice Hall, 2008. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.

The cdh topology is the smallest Grothendieck topology whose covering morphisms include those of the proper cdh topology and those of the Nisnevich topology.

The logical topology can be considered isomorphic to the physical topology, as vice versa. Early twisted pair Ethernet with a single hub is a logical bus topology with a physical star topology. While token ring is a logical ring topology with a physical star topology.

In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on X that makes those functions continuous. The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these. The dual notion is the final topology, which for a given family of functions mapping to a set X is the finest topology on X that makes those functions continuous.

The v-topology (or universally subtrusive topology) is equivalent to the h-topology on Noetherian schemes. On more general schemes, the v-topology has more covers.

In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.

The topology generated by these bases is the relative topology. The conclusion is that the relative topology is the same as the group topology. Next, construct coordinate charts on . First define .

The topology generated by any subset (including by the empty set ) is equal to the trivial topology }. If is a topology on and is a basis for then the topology generated by is . Thus any basis for a topology is also a subbasis for . If is any subset of then the topology generated by will be a subset of .

The topology on this space is called the graph topology.

The Helly space is a subset of II. The space II has its own topology, namely the product topology. The Helly space has a topology; namely the induced topology as a subset of II.

In mathematics, pointless topology (also called point-free or pointfree topology, or locale theory) is an approach to topology that avoids mentioning points.

In functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual. In other words the Mackey topology does not make linear functions continuous which were discontinuous in the default topology. A topological vector space (TVS) is called a Mackey space if its topology is the same as the Mackey topology. The Mackey topology is the opposite of the weak topology, which is the coarsest topology on a topological vector space which preserves the continuity of all linear functions in the continuous dual.

A physical extended star topology in which repeaters are replaced with hubs or switches is a type of hybrid network topology and is referred to as a physical hierarchical star topology, although some texts make no distinction between the two topologies. A physical hierarchical star topology can also be referred as a tier-star topology, this topology differs from a tree topology in the way star networks are connected together. A tier-star topology uses a central node, while a tree topology uses a central bus and can also be referred as a star-bus network.

Computable topology is a discipline in mathematics that studies the topological and algebraic structure of computation. Computable topology is not to be confused with algorithmic or computational topology, which studies the application of computation to topology.

In topology, a branch of mathematics, an extension topology is a topology placed on the disjoint union of a topological space and another set. There are various types of extension topology, described in the sections below.

In topology and related areas of mathematics, an induced topology on a topological space is a topology that makes a given (inducing) function or collection of functions continuous from this topological space. A coinduced topology or final topology makes the given (coinducing) collection of functions continuous to this topological space.

In mathematics, especially in algebraic geometry, the v-topology (also known as the universally subtrusive topology) is a Grothendieck topology whose covers are characterized by lifting maps from valuation rings. This topology was introduced by and studied further by , who introduced the name v-topology, where v stands for valuation.

In general topology, a branch of mathematics, the integer broom topology, is an example of a topology on the so-called integer broom space X.

The Mackey–Arens theorem, named after George Mackey and Richard Arens, characterizes all possible dual topologies on a locally convex space. The theorem shows that the coarsest dual topology is the weak topology, the topology of uniform convergence on all finite subsets of X', and the finest topology is the Mackey topology, the topology of uniform convergence on all absolutely convex weakly compact subsets of X'.

Cauer topology is usually thought of as an unbalanced ladder topology. A ladder network consists of cascaded asymmetrical L-sections (unbalanced) or C-sections (balanced). In low pass form the topology would consist of series inductors and shunt capacitors. Other bandforms would have an equally simple topology transformed from the lowpass topology.

There are several different filter topologies available to implement a linear analogue filter. The most often used topology for a passive realisation is Cauer topology and the most often used topology for an active realisation is Sallen–Key topology.

The topology on is induced by a translation-invariant metric on . 5. The topology on is induced by an -norm. 6. The topology on is induced by a monotone -norm. 7. The topology on is induced by a total paranorm.

In mathematics and theoretical computer science the Lawson topology, named after Jimmie D. Lawson, is a topology on partially ordered sets used in the study of domain theory. The lower topology on a poset P is generated by the subbasis consisting of all complements of principal filters on P. The Lawson topology on P is the smallest common refinement of the lower topology and the Scott topology on P.

In functional analysis and related areas of mathematics the strong topology on the continuous dual space of a topological vector space (TVS) is the finest polar topology, the topology with the most open sets, on a dual pair. The coarsest polar topology is called weak topology. When the continuous dual space of a TVS is endowed with the this topology then it is called the strong dual space of .

In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set X, with respect to a family of functions into X, is the finest topology on X that makes those functions continuous. The dual notion is the initial topology, which for a given family of functions from a set X is the coarsest topology on X that makes those functions continuous.

The finest topology on X is the discrete topology; this topology makes all subsets open. The coarsest topology on X is the trivial topology; this topology only admits the empty set and the whole space as open sets. In function spaces and spaces of measures there are often a number of possible topologies. See topologies on the set of operators on a Hilbert space for some intricate relationships.

3-manifold theory is considered a part of low-dimensional topology or geometric topology.

However, a dcpo equipped with the Scott topology need not be sober: the specialization order induced by the topology of a sober space makes that space into a dcpo, but the Scott topology derived from this order is finer than the original topology.

In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale topology was originally introduced by Grothendieck to define étale cohomology, and this is still the étale topology's most well- known use.

In general topology, a branch of mathematics, the Appert topology, named for , is a topology on the set } of positive integers. In the Appert topology, the open sets are those that do not contain 1, and those that asymptotically contain almost every positive integer. The space X with the Appert topology is called the Appert space.

While defined on all schemes, the h and qfh topology are only ever used on Noetherian schemes. The h topology has various non-equivalent extensions to non-Noetherian schemes including the ph topologyA cohomological bound for the h-topology and the v topology. The proper cdh topology is defined as follows. Let be a proper morphism.

Let be a real or complex vector space. ;Trivial topology The trivial topology (or the indiscrete topology) } is always a TVS topology on any vector space , so it is obviously the coarsest TVS topology possible. This simple observation allows us to conclude that the intersection of any collection of TVS topologies on always contains a TVS topology. Any vector space (including those that are infinite dimensional) endowed with the trivial topology is a compact (and thus locally compact) complete pseudometrizable seminormable locally convex topological vector space.

Then the set } forms a neighborhood basisNote that each set is a neighborhood of the origin for this topology, but it is not necessarily an open neighborhood of the origin. at the origin for a unique translation- invariant topology on , where this topology is not necessarily a vector topology (i.e. it might not make into a TVS). This topology does not depend on the neighborhood basis that was chosen and it is known as the topology of uniform convergence on the sets in or as the -topology.

The topology on is induced by a translation-invariant pseudometric on . 4. The topology on is induced by an -seminorm. 5. The topology on is induced by a paranorm.

Topology was a peer-reviewed mathematical journal covering topology and geometry. It was established in 1962 and was published by Elsevier. The last issue of Topology appeared in 2009.

Examples of linear, pericyclic, and coarctate transition states In the classification of organic reactions by transition state topology, a coarctate reaction (from L. coarctare "to constrict") is a third, comparatively uncommon topology, after linear topology and pericyclic topology (itself subdivided into Hückel and Möbius topologies).

If is given the discrete topology, and if is given the product topology, and is viewed as a subspace of and is given the subspace topology, then acts densely on if and only if is dense set in with this topology.It turns out this topology is the same as the compact-open topology in this case. Herstein, p. 41 uses this description.

In some ways, the opposite of the discrete topology is the trivial topology (also called the indiscrete topology), which has the fewest possible open sets (just the empty set and the space itself). Where the discrete topology is initial or free, the indiscrete topology is final or cofree: every function from a topological space to an indiscrete space is continuous, etc.

The continuous dual space of (Y,\tau(Y,X,b)) is (in the exact same way as was described for the weak topology). Moreover, the Mackey topology is the finest locally convex topology on for which this is true, which is what makes this topology important. Since in general, the convex balanced hull of a \sigma(Y,X,b)-compact subset of need not be \sigma(Y,X,b)-compact, the Mackey topology may be strictly coarser than the topology c(X,Y,b). Since every \sigma(Y,X,b)-compact set is \sigma(Y,X,b)-bounded, the Mackey topology is coarser than the strong topology b(X,Y,b).

Let be any non-empty open subset of (e.g. could be a non-empty bounded open interval in ) and let denote the subspace topology on that inherits from (so ). Then the topology generated by on is equal to the union (see this footnote for an explanation),Since is a topology on and is an open subset of , it is easy to verify that is a topology on . Since isn't a topology on , is clearly the smallest topology on containing ).

The Alexandrov topology on spacetime, is the coarsest topology such that both Y^+(E) and Y^-(E) are open for all subsets E \subset M. Here the base of open sets for the topology are sets of the form Y^+(x) \cap Y^-(y) for some points \,x,y \in M. This topology coincides with the manifold topology if and only if the manifold is strongly causal but it is coarser in general. Note that in mathematics, an Alexandrov topology on a partial order is usually taken to be the coarsest topology in which only the upper sets Y^+(E) are required to be open. This topology goes back to Pavel Alexandrov. Nowadays, the correct mathematical term for the Alexandrov topology on spacetime (which goes back to Alexandr D. Alexandrov) would be the interval topology, but when Kronheimer and Penrose introduced the term this difference in nomenclature was not as clear, and in physics the term Alexandrov topology remains in use.

Locales and frames form the foundation of pointless topology, which, instead of building on point-set topology, recasts the ideas of general topology in categorical terms, as statements on frames and locales.

In mathematics, the overlapping interval topology is a topology which is used to illustrate various topological principles.

This follows directly from the weak operator topology being coarser than the strong operator topology: for every point in , every open neighborhood of in the weak operator topology is also open in the strong operator topology and therefore contains a member of ; therefore is also a member of .

The Minkowski functional of the disk in guarantees that is well-defined and forms a seminorm on . The locally convex topology topology induced by this seminorm is the topology that was defined before.

The Appert topology is closely related to the Fort space topology that arises from giving the set of integers greater than one the discrete topology, and then taking the point 1 as the point at infinity in a one point compactification of the space. The Appert topology is finer than the Fort space topology, as any cofinite subset of X has asymptotic density equal to 1.

Figure 1.12 Ladder topology can be extended without limit and is much used in filter designs. There are many variations on ladder topology, some of which are discussed in the Electronic filter topology and Composite image filter articles. Figure 1.13. Anti-ladder topology The balanced form of ladder topology can be viewed as being the graph of the side of a prism of arbitrary order.

Similarly, if then we have the dual definition of the weak topology on induced by (and ), which is denoted by or simply (see footnote for details).The weak topology on is the weakest TVS topology on making all maps continuous, as ranges over . We also use the dual notation of , , or simply to denote endowed with the weak topology . If we do not indicate what the subset is, then by the weak topology on we mean the weak topology on induced by .

The double-pointed cofinite topology is the cofinite topology with every point doubled; that is, it is the topological product of the cofinite topology with the indiscrete topology on a two-element set. It is not T0 or T1, since the points of the doublet are topologically indistinguishable. It is, however, R0 since the topologically distinguishable points are separable. An example of a countable double-pointed cofinite topology is the set of even and odd integers, with a topology that groups them together.

Figure 2. The Akerberg-Mossberg biquad filter topology. Figure 2 shows a variant of the Tow-Thomas topology, known as Akerberg-Mossberg topology, that uses an actively compensated Miller integrator, which improves filter performance.

In mathematics, a Lawvere–Tierney topology is an analog of a Grothendieck topology for an arbitrary topos, used to construct a topos of sheaves. A Lawvere–Tierney topology is also sometimes also called a local operator or coverage or topology or geometric modality. They were introduced by and Myles Tierney.

Topology of a protein can be used to classify proteins as well. Knot theory and circuit topology are two topology frameworks developed for classification of protein folds based on chain crossing and intrachain contacts respectively.

More precisely, it is an adaptation of the Fell topology construction, which itself derives from the Vietoris topology concept.

Ralph Louis Cohen Ralph Louis Cohen (born 1952) is an American mathematician, specializing in algebraic topology and differential topology.

Declaring the complements of the closed sets to be open, this defines a topology (the Zariski topology) on CPn.

While the box topology has a somewhat more intuitive definition than the product topology, it satisfies fewer desirable properties. In particular, if all the component spaces are compact, the box topology on their Cartesian product will not necessarily be compact, although the product topology on their Cartesian product will always be compact. In general, the box topology is finer than the product topology, although the two agree in the case of finite direct products (or when all but finitely many of the factors are trivial).

In algebra, a linear topology on a left A-module M is a topology on M that is invariant under translations and admits a fundamental system of neighborhood of 0 that consists of submodules of M. If there is such a topology, M is said to be linearly topologized. If A is given a discrete topology, then M becomes a topological A-module with respect to a linear topology.

Colin Rourke (born 1 January 1943) is a British mathematician, who has published papers in PL topology, low-dimensional topology, differential topology, group theory, relativity and cosmology. He is an emeritus professor at the Mathematics Institute of the University of Warwick and a founding editor of the journals Geometry & Topology and Algebraic & Geometric Topology, published by Mathematical Sciences Publishers, where he is the vice chair of its board of directors.

Geometry has local structure (or infinitesimal), while topology only has global structure. Alternatively, geometry has continuous moduli, while topology has discrete moduli. By examples, an example of geometry is Riemannian geometry, while an example of topology is homotopy theory. The study of metric spaces is geometry, the study of topological spaces is topology.

This type of six-bar linkage is said to have the Stephenson topology. The Klann linkage has the Stephenson topology.

There is a similar definition for Nisnevich sheaf with transfers, where the Etale topology is switched with the Nisnevich topology.

All possible polar topologies on a dual pair are finer than the weak topology and coarser than the strong topology.

The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology.Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.Adams, Colin Conrad, and Robert David Franzosa.

An important basic result states that the Gromov topology, the weak topology and the length function topology on Xn coincide.Vincent Guirardel, Gilbert Levitt, Deformation spaces of trees. Groups, Geometry, and Dynamics 1 (2007), no. 2, 135–181.

The topological structure of (called standard topology, Euclidean topology, or usual topology) can be obtained not only from Cartesian product. It is also identical to the natural topology induced by Euclidean metric discussed above: a set is open in the Euclidean topology if and only if it contains an open ball around each of its points. Also, is a linear topological space (see continuity of linear maps above), and there is only one possible (non-trivial) topology compatible with its linear structure. As there are many open linear maps from to itself which are not isometries, there can be many Euclidean structures on which correspond to the same topology.

The real numbers form a metric space: the distance between x and y is defined as the absolute value . By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology—in the order topology as ordered intervals, in the metric topology as epsilon-balls. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. The reals are a contractible (hence connected and simply connected), separable and complete metric space of Hausdorff dimension 1\.

Observe that the weak topology depends entirely on the function and the usual topology on (the topology doesn't even depend on the algebraic structures of and ). :Definition and notation: If we attach "" to a topological definition (e.g. -converges, -bounded, , etc.) then we mean that definition when the first space (i.e. X) carries the topology.

Below are two cladograms showing the possible genera that are included within the ornithocheirid family. The cladogram to the left is a topology recovered by Jacobs and colleagues, and the one to the right is a topology recovered by Pentland and colleagues. Topology 1: Jacobs et al. (2019). Topology 2: Pentland et al. (2019).

Definition:Luca Bombelli website The topology \rho in which a subset E \subset M is open if for every timelike curve c there is a set O in the manifold topology such that E \cap c = O \cap c. It is the finest topology which induces the same topology as M does on timelike curves.

In several mathematical areas, including harmonic analysis, topology, and number theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers (equipped with the discrete topology), or the real numbers or the circle (both with their usual topology) are locally compact abelian groups.

The cocountable extension topology is the topology on the real line generated by the union of the usual Euclidean topology and the cocountable topology. Sets are open in this topology if and only if they are of the form U \ A where U is open in the Euclidean topology and A is countable. This space is completely Hausdorff and Urysohn, but not regular (and thus not Tychonoff). There exist spaces which are Hausdorff but not Urysohn, and spaces which are Urysohn but not completely Hausdorff or regular Hausdorff.

In contrast, logical topology is the way that the signals act on the network media, or the way that the data passes through the network from one device to the next without regard to the physical interconnection of the devices. A network's logical topology is not necessarily the same as its physical topology. For example, the original twisted pair Ethernet using repeater hubs was a logical bus topology carried on a physical star topology. Token ring is a logical ring topology, but is wired as a physical star from the media access unit.

Suppose that is a pairing of vector spaces over . If then the weak topology on induced by (and ) is the weakest TVS topology on , denoted by or simply , making all maps continuous, as ranges over . We use , , or (if no confusion could arise) simply to denote endowed with the weak topology . If we do not indicate what the subset is, then by the weak topology on we mean the weak topology on induced by .

Any ordinal number can be made into a topological space by endowing it with the order topology; this topology is discrete if and only if the ordinal is a countable cardinal, i.e. at most ω. A subset of ω + 1 is open in the order topology if and only if either it is cofinite or it does not contain ω as an element. See the Topology and ordinals section of the "Order topology" article.

In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any finite family of open sets is open; in Alexandrov topologies the finite restriction is dropped. A set together with an Alexandrov topology is known as an Alexandrov-discrete space or finitely generated space. Alexandrov topologies are uniquely determined by their specialization preorders.

Throughout this section, T will denote the K-topology and (R, T) will denote the set of all real numbers with the K-topology as a topological space. 1\. The topology T on R is strictly finer than the standard topology on R but not comparable with the lower limit topology on R 2\. From the previous example, it follows that (R, T) is not compact 3\. (R, T) is Hausdorff but not regular.

If Y is a subset of X, then Y inherits a total order from X. The set Y therefore has an order topology, the induced order topology. As a subset of X, Y also has a subspace topology. The subspace topology is always at least as fine as the induced order topology, but they are not in general the same. For example, consider the subset Y = {–1} ∪ {1/n}n∈N in the rationals.

In functional analysis and related areas of mathematics a polar topology, topology of -convergence or topology of uniform convergence on the sets of is a method to define locally convex topologies on the vector spaces of a pairing.

The fine topology on the Euclidean space \R^n is defined to be the coarsest topology making all subharmonic functions (equivalently all superharmonic functions) continuous. Concepts in the fine topology are normally prefixed with the word 'fine' to distinguish them from the corresponding concepts in the usual topology, as for example 'fine neighbourhood' or 'fine continuous'.

In general topology, a branch of mathematics, the evenly spaced integer topology is the topology on the set of integers } generated by the family of all arithmetic progressions. It is a special case of the profinite topology on a group. This particular topological space was introduced by where it was used to prove the infinitude of primes.

The h topology combines a number of useful properties of its various "sub"topologies. Since if is finer than the Zariski topology, h-locally every scheme is affine. Since it is finer than the Nisnevich_topology, h-locally regular immersions look like zero sections of vector bundles. It is also finer than the étale topology and the fppf topology.

In general, the product of the topologies of each Xi forms a basis for what is called the box topology on X. In general, the box topology is finer than the product topology, but for finite products they coincide.

The Mackey–Arens theorem states that all possible dual topologies are finer than the weak topology and coarser than the Mackey topology.

In mathematics, the geometric topology is a topology one can put on the set H of hyperbolic 3-manifolds of finite volume.

In mathematics, particularly in functional analysis, a Mackey space is a locally convex topological vector space X such that the topology of X coincides with the Mackey topology τ(X,X′), the finest topology which still preserves the continuous dual.

In algebraic geometry, the syntomic topology is a Grothendieck topology introduced by . Mazur defined a morphism to be syntomic if it is flat and locally a complete intersection. The syntomic topology is generated by surjective syntomic morphisms of affine schemes.

In some contexts, it is helpful to place other topologies on the set of real numbers, such as the lower limit topology or the Zariski topology. For the real numbers, the latter is the same as the finite complement topology.

The following is one of the most important theorems in duality theory. It follows that the Mackey topology , which recall is the polar topology generated by all -compact disks in , is the strongest locally convex topology on that is compatible with the pairing . A locally convex space whose given topology is identical to the Mackey topology is called a Mackey space. The following consequence of the above Mackey-Arens theorem is also called the Mackey-Arens theorem.

Figure 1.8 Bridge topology is an important topology with many uses in both linear and non-linear applications, including, amongst many others, the bridge rectifier, the Wheatstone bridge and the lattice phase equaliser. There are several ways that bridge topology is rendered in circuit diagrams. The first rendering in figure 1.8 is the traditional depiction of a bridge circuit. The second rendering clearly shows the equivalence between the bridge topology and a topology derived by series and parallel combinations.

The usual topology on the real numbers has a subbase consisting of all semi-infinite open intervals either of the form or , where and are real numbers. Together, these generate the usual topology, since the intersections for generate the usual topology. A second subbase is formed by taking the subfamily where and are rational. The second subbase generates the usual topology as well, since the open intervals with , rational, are a basis for the usual Euclidean topology.

Let R be a commutative ring, and I an ideal of R. Given an R-module M, the sequence I^n M of submodules of M forms a filtration of M. The I-adic topology on M is then the topology associated to this filtration. If M is just the ring R itself, we have defined the I-adic topology on R. When R is given the I-adic topology, R becomes a topological ring. If an R-module M is then given the I-adic topology, it becomes a topological R-module, relative to the topology given on R.

For this reason when considering the topology of computation it is common to focus on the topology of λ-calculus. Note that this is not necessarily a complete description of the topology of computation, since functions which are equivalent in the Church-Turing sense may still have different topologies. The topology of λ-calculus is the Scott topology, and when restricted to continuous functions the type free λ-calculus amounts to a topological space reliant on the tree topology. Both the Scott and Tree topologies exhibit continuity with respect to the binary operators of application ( f applied to a = fa ) and abstraction ((λx.

Topology selection plays an important role in routing because the network topology decides the transmission path of the data packets to reach the proper destination. Here, all the topologies (Flat / Unstructured, cluster, tree, chain and hybrid topology) are not feasible for reliable data transmission on sensor nodes mobility. Instead of single topology, hybrid topology plays a vital role in data collection, and the performance is good. Hybrid topology management schemes include the Cluster Independent Data Collection Tree (CIDT).R. Velmani, and B. Kaarthick, 2014. An Energy Efficient Data Gathering in Dense Mobile Wireless Sensor Networks,” ISRN Sensor Networks, vol.

The Zariski topology in the set-theoretic sense is then replaced by a Zariski topology in the sense of Grothendieck topology. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely the étale topology, and the two flat Grothendieck topologies: fppf and fpqc. Nowadays some other examples have become prominent, including the Nisnevich topology. Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions, leading to Artin stacks and, even finer, Deligne–Mumford stacks, both often called algebraic stacks.

It is Hausdorff if and only if . ;Finest vector topology There exists a TVS topology on that is finer than every other TVS- topology on (that is, any TVS-topology on is necessarily a subset of ). Every linear map from into another TVS is necessarily continuous. If has an uncountable Hamel basis then is not locally convex and not metrizable.

Figure 1.10. Bridged T topology Bridged T topology is derived from bridge topology in a way explained in the Zobel network article. There are many derivative topologies also discussed in the same article. Figure 1.11 There is also a twin-T topology which has practical applications where it is desirable to have the input and output share a common (ground) terminal.

Although topology is the field of mathematics that formalizes and generalizes the intuitive notion of "continuous deformation" of objects, it gives rise to many discrete topics; this can be attributed in part to the focus on topological invariants, which themselves usually take discrete values. See combinatorial topology, topological graph theory, topological combinatorics, computational topology, discrete topological space, finite topological space, topology (chemistry).

This is the smallest T1 topology on any infinite set. Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations. The real line can also be given the lower limit topology.

Low-dropout (LDO) regulators work in the same way as all linear voltage regulators. The main difference between LDO and non-LDO regulators is their schematic topology. Instead of an emitter follower topology, low-dropout regulators use open collector or open drain topology. In this topology, the transistor may be easily driven into saturation with the voltages available to the regulator.

Visualization of an all-to-all algorithm in a ring topology. Visualization of an all-to-all algorithm in a mesh topology. We consider a single-ported machine. The way the data is routed through the network depends on its underlying topology.

World Wide Web topology is the network topology of the World Wide Web, as seen as a network of web pages connected by hyperlinks. The Jellyfish and Bow Tie models are two attempts at modeling the topology of hyperlinks between web pages.

Topology optimization is a type of structural optimization technique which can optimize material layout within a given design space. Compared to other typical structural optimization techniques, such as size optimization or shape optimization, topology optimization can update both shape and topology of a part. However, the complex optimized shapes obtained from topology optimization are always difficult to handle for traditional manufacturing processes such as CNC machining. To solve this issue, additive manufacturing processes can be applied to fabricate topology optimization result.

Considered as its functor of points, a scheme is a functor which is a sheaf of sets for the Zariski topology on the category of commutative rings, and which, locally in the Zariski topology, is an affine scheme. This can be generalized in several ways. One is to use the étale topology. Michael Artin defined an algebraic space as a functor which is a sheaf in the étale topology and which, locally in the étale topology, is an affine scheme.

Lattice topology X-section phase correction filter Both the T-section (from ladder topology) and the bridge-T (from Zobel topology) can be transformed into a lattice topology filter section but in both cases this results in high component count and complexity. The most common application of lattice filters (X-sections) is in all-pass filters used for phase equalisation.Zobel, 1931 Although T and bridged-T sections can always be transformed into X-sections the reverse is not always possible because of the possibility of negative values of inductance and capacitance arising in the transform. Lattice topology is identical to the more familiar bridge topology, the difference being merely the drawn representation on the page rather than any real difference in topology, circuitry or function.

The Sallen–Key topology is an electronic filter topology used to implement second-order active filters that is particularly valued for its simplicity."EE315A Course Notes - Chapter 2"-B. Murmann It is a degenerate form of a voltage-controlled voltage-source (VCVS) filter topology.

The curvature of the universe places constraints on the topology. If the spatial geometry is spherical, i.e., possess positive curvature, the topology is compact. For a flat (zero curvature) or a hyperbolic (negative curvature) spatial geometry, the topology can be either compact or infinite.

Ryszard Engelking (born 1935 in Sosnowiec) is a Polish mathematician. He was working mainly on general topology and dimension theory. He is author of several influential monographs in this field. The 1989 edition of his General Topology is nowadays a standard reference for Topology.

Pavel Sergeyevich Alexandrov (), sometimes romanized Paul Alexandroff (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote about three hundred papers, making important contributions to set theory and topology. In topology, the Alexandroff compactification and the Alexandrov topology are named after him.

If is a vector subspace of the algebraic dual space of then we will assume that they are associated with the canonical pairing . In this case, the weak topology on (resp. the weak topology on ), denoted by (resp. by ) is the weak topology on (resp.

The space C of continuous functions on E is a subspace of D. The Skorokhod topology relativized to C coincides with the uniform topology there.

To prove that Tychonoff's theorem in its general version implies the axiom of choice, we establish that every infinite cartesian product of non-empty sets is nonempty. The trickiest part of the proof is introducing the right topology. The right topology, as it turns out, is the cofinite topology with a small twist. It turns out that every set given this topology automatically becomes a compact space.

In category theory, a branch of mathematics, a sieve is a way of choosing arrows with a common codomain. It is a categorical analogue of a collection of open subsets of a fixed open set in topology. In a Grothendieck topology, certain sieves become categorical analogues of open covers in topology. Sieves were introduced by in order to reformulate the notion of a Grothendieck topology.

The study of topology in mathematics extends all over through point set topology, algebraic topology, differential topology, and all the related paraphernalia, such as homology theory, homotopy theory. The development of abstract algebra brought with itself group theory, rings, fields, and Galois theory. This list could be expanded to include most fields of mathematics, including measure theory, ergodic theory, probability, representation theory, and differential geometry.

Let be a topological space with topology . A subbase of is usually defined as a subcollection of satisfying one of the two following equivalent conditions: #The subcollection generates the topology . This means that is the smallest topology containing : any topology on containing must also contain . #The collection of open sets consisting of all finite intersections of elements of , together with the set , forms a basis for .

The star topology is considered the easiest topology to design and implement. One advantage of the star topology is the simplicity of adding additional nodes. The primary disadvantage of the star topology is that the hub represents a single point of failure. Also, since all peripheral communication must flow through the central hub, the aggregate central bandwidth forms a network bottleneck for large clusters.

This may be, for instance, because the input and output connections are made with co-axial topology. Connecting together an input and output terminal is not allowable with normal bridge topology and for this reason Twin-T is used where a bridge would otherwise be used for balance or null measurement applications. The topology is also used in the twin-T oscillator as a sine wave generator. The lower part of figure 1.11 shows twin-T topology redrawn to emphasise the connection with bridge topology.

If X is a regularly ordered vector lattice then the ordered topology is the finest locally convex TVS topology on X making X into a locally convex vector lattice. If in addition X is order complete then X with the order topology is a barreled space and every band decomposition of X is a topological direct sum for this topology. In particular, if the order of a vector lattice X is regular then the order topology is generated by the family of all lattice seminorms on X.

Geometric topology is a branch of topology that primarily focuses on low- dimensional manifolds (that is, spaces of dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology.R.B. Sher and R.J. Daverman (2002), Handbook of Geometric Topology, North-Holland. Some examples of topics in geometric topology are orientability, handle decompositions, local flatness, crumpling and the planar and higher-dimensional Schönflies theorem. In high-dimensional topology, characteristic classes are a basic invariant, and surgery theory is a key theory.

The subbase consisting of all semi-infinite open intervals of the form alone, where is a real number, does not generate the usual topology. The resulting topology does not satisfy the T1 separation axiom, since all open sets have a non-empty intersection. The initial topology on defined by a family of functions , where each has a topology, is the coarsest topology on such that each is continuous. Because continuity can be defined in terms of the inverse images of open sets, this means that the initial topology on is given by taking all , where ranges over all open subsets of , as a subbasis.

Emmy Murphy is an American mathematician and an associate professor at Northwestern University who works in the area of symplectic topology, contact geometry and geometric topology.

For example, some derivations require a fixed choice of the topology, while any consistent quantum theory of gravity should include topology change as a dynamical process.

In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space (X, ≤) is the finest locally convex topological vector space (TVS) topology on X for which every order interval is bounded, where an order interval in X is a set of the form [a, b] := { z ∈ X : a ≤ z and z ≤ b } where a and b belong to X. The order topology is an important topology that is used frequently in the theory of ordered topological vector spaces because the topology stems directly from the algebraic and order theoretic properties of (X, ≤), rather than from some topology that X starts out having. This allows for establishing intimate connections between this topology and the algebraic and order theoretic properties of (X, ≤). For many ordered topological vector spaces that occur in analysis, their topologies are identical to the order topology.

The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of the variety. In the case of an algebraic variety over the complex numbers, the Zariski topology is thus coarser than the usual topology, as every algebraic set is closed for the usual topology. The generalization of the Zariski topology to the set of prime ideals of a commutative ring follows from Hilbert's Nullstellensatz, that establishes a bijective correspondence between the points of an affine variety defined over an algebraically closed field and the maximal ideals of the ring of its regular functions. This suggests defining the Zariski topology on the set of the maximal ideals of a commutative ring as the topology such that a set of maximal ideals is closed if and only if it is the set of all maximal ideals that contain a given ideal.

The submanifold topology on an immersed submanifold need not be the relative topology inherited from M. In general, it will be finer than the subspace topology (i.e. have more open sets). Immersed submanifolds occur in the theory of Lie groups where Lie subgroups are naturally immersed submanifolds.

This would form a robust irreversible switch. There is no one-to-one correspondence between network topology, since many networks have a similar input and output relationship. A network topology does not imply input or output, and similarly input or output does not imply network topology.

More generally, the Euclidean spaces Rn can be given a topology. In the usual topology on Rn the basic open sets are the open balls. Similarly, C, the set of complex numbers, and Cn have a standard topology in which the basic open sets are open balls.

Topology of a bus network A bus network is a network topology in which nodes are directly connected to a common half-duplex link called a bus.

In mathematics, the algebraic topology on the set of group representations from G to a topological group H is the topology of pointwise convergence, i.e. pi converges to p if the limit of pi(g) = p(g) for every g in G. This terminology is often used in the case of the algebraic topology on the set of discrete, faithful representations of a Kleinian group into PSL(2,C). Another topology, the geometric topology (also called the Chabauty topology), can be put on the set of images of the representations, and its closure can include extra Kleinian groups that are not images of points in the closure in the algebraic topology. This fundamental distinction is behind the phenomenon of hyperbolic Dehn surgery and plays an important role in the general theory of hyperbolic 3-manifolds.

A Cartesian product of a family of topological vector spaces, when endowed with the product topology, is a topological vector space. For instance, the set of all functions : this set can be identified with the product space and carries a natural product topology. With this topology, becomes a topological vector space, endowed with a topology called the topology of pointwise convergence. The reason for this name is the following: if is a sequence of elements in , then has limit if and only if has limit for every real number x.

The topology of an electronic circuit is the form taken by the network of interconnections of the circuit components. Different specific values or ratings of the components are regarded as being the same topology. Topology is not concerned with the physical layout of components in a circuit, nor with their positions on a circuit diagram; similarly to the mathematic concept of topology, it is only concerned with what connections exist between the components. There may be numerous physical layouts and circuit diagrams that all amount to the same topology.

If one starts with the standard topology on the real line R and defines a topology on the product of n copies of R in this fashion, one obtains the ordinary Euclidean topology on Rn. The Cantor set is homeomorphic to the product of countably many copies of the discrete space {0,1} and the space of irrational numbers is homeomorphic to the product of countably many copies of the natural numbers, where again each copy carries the discrete topology. Several additional examples are given in the article on the initial topology.

A torus, one of the most frequently studied objects in algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.

A fully connected network, complete topology, or full mesh topology is a network topology in which there is a direct link between all pairs of nodes. In a fully connected network with n nodes, there are n(n-1)/2 direct links. Networks designed with this topology are usually very expensive to set up, but provide a high degree of reliability due to the multiple paths for data that are provided by the large number of redundant links between nodes. This topology is mostly seen in military applications.

Let be a non-trivial (i.e. }) real or complex vector space and let be the translation-invariant trivial metric on defined by and for all such that . The topology that induces on is the discrete topology, which makes into a commutative topological group under addition but does not form a vector topology on because is disconnected but every vector topology is connected. What fails is that scalar multiplication isn't continuous on .

In mathematics, and especially general topology, the interlocking interval topology is an example of a topology on the set , i.e. the set of all positive real numbers that are not positive whole numbers.Steen & Seebach (1978) pp.77 – 78 To give the set S a topology means to say which subsets of S are "open", and to do so in a way that the following axioms are met:Steen & Seebach (1978) p.

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. The discrete topology is the finest topology that can be given on a set, i.e., it defines all subsets as open sets. In particular, each singleton is an open set in the discrete topology.

Logical topology is the arrangement of devices on a computer network and how they communicate with one another. Logical topologies describe how signals act on the network. In contrast, a physical topology defines how nodes in a network are physically linked and includes aspects such as geographic location of nodes and physical distances between nodes. The logical topology defines how nodes in a network communicate across its physical topology.

In general, however, there is no unique subbasis for a given topology. Thus, we can start with a fixed topology and find subbases for that topology, and we can also start with an arbitrary subcollection of the power set and form the topology generated by that subcollection. We can freely use either equivalent definition above; indeed, in many cases, one of the two conditions is more useful than the other.

He was also a long-time member of the editorial boards of two mathematics journals, the Pacific Journal of Mathematics and Topology and its Applications. Dugundji is the author of the textbook Topology (Allyn and Bacon, 1966), on general topology. Reviewer M. Edelstein wrote that this was "one of the best among the numerous books on the subject",Review of Topology by M. Edelstein, . and it went through numerous reprintings.

Henselian rings are the local rings of "points" with respect to the Nisnevich topology, so the spectra of these rings do not admit non-trivial connected coverings with respect to the Nisnevich topology. Likewise strict Henselian rings are the local rings of geometric points in the étale topology.

Because of the incompatibility of the profinite topology on GK and the usual (Euclidean) topology on complex vector spaces, the image of an Artin representation is always finite.

If M is a solid vector subspace of a vector lattice X, then the order topology of X/M is the quotient of the order topology on X.

The topology on this space is essentially the weak topology, the open sets being the cylinder sets. An -dimensional superspace is just the -fold product of exterior algebras.

In this language, the definition of the étale topology is succinct but abstract: It is the topology generated by the pretopology whose covering families are jointly surjective families of étale morphisms. The small étale site of X is the category O(Xét) whose objects are schemes U with a fixed étale morphism U → X. The morphisms are morphisms of schemes compatible with the fixed maps to X. The big étale site of X is the category Ét/X, that is, the category of schemes with a fixed map to X, considered with the étale topology. The étale topology can be defined using slightly less data. First, notice that the étale topology is finer than the Zariski topology.

In functional analysis and related areas of mathematics a dual topology is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space. The different dual topologies for a given dual pair are characterized by the Mackey–Arens theorem. All locally convex topologies with their continuous dual are trivially a dual pair and the locally convex topology is a dual topology. Several topological properties depend only on the dual pair and not on the chosen dual topology and thus it is often possible to substitute a complicated dual topology by a simpler one.

Every group G may be made into a topological group by taking as a basis of open neighbourhoods of the identity, the collection of all normal subgroups of finite index in G. The resulting topology is called the profinite topology on G. A group is residually finite if, and only if, its profinite topology is Hausdorff. A group whose cyclic subgroups are closed in the profinite topology is said to be \Pi_C\,. Groups, each of whose finitely generated subgroups are closed in the profinite topology are called subgroup separable (also LERF, for locally extended residually finite). A group in which every conjugacy class is closed in the profinite topology is called conjugacy separable.

Equivalently, it is a surjective TVS embedding. Many properties of TVSs that are studied, such as local convexity, metrizability, completeness, and normability, are invariant under TVS isomorphisms. ;A necessary condition for a vector topology All of the above conditions are consequently a necessity for a topology to form a vector topology.

Splitting a necklace with two cuts. Combinatorial analogs of concepts and methods in topology are used to study graph coloring, fair division, partitions, partially ordered sets, decision trees, necklace problems and discrete Morse theory. It should not be confused with combinatorial topology which is an older name for algebraic topology.

In topology, a subbase (or subbasis) for a topological space with topology is a subcollection of that generates , in the sense that is the smallest topology containing . A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below.

Circuit topology relations in a chain with two binary contacts. The circuit topology of a linear polymer refers to arrangement of its intra-molecular contacts. Examples of linear polymers with intra-molecular contacts are nucleic acids and proteins. For defining the circuit topology, contacts are defined depending on the context.

Circuit topology along with contact order and size are determinants of folding rate of linear polymers. The topology of the cellular proteome and natural RNA reflect evolutionary constraints on biomolecular structures. Topology landscape of biomolecules can be characterized and evolution of molecules can be studied as transition pathways within the landscape.

Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. It establishes an analogy between number fields and closed, orientable 3-manifolds.

The structural topology is the same, consisting of a tiered star topology with a root hub at level 0 and hubs at lower levels to provide bus connectivity to devices.

Starting with only the weak topology, we may obtain a range of locally convex topologies by using polar sets. Such topologies are called polar topologies. The weak topology is the weakest topology of this range. Throughout, will be a pairing over and 𝒢 will be a non-empty collection of -bounded subsets of .

Let C be any category. To define the discrete topology, we declare all sieves to be covering sieves. If C has all fibered products, this is equivalent to declaring all families to be covering families. To define the indiscrete topology, also known as the coarse or chaotic topology,SGA IV, II 1.1.4.

A given set may have many different topologies. If a set is given a different topology, it is viewed as a different topological space. Any set can be given the discrete topology in which every subset is open. The only convergent sequences or nets in this topology are those that are eventually constant.

In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the theory of motives. It was originally introduced by Yevsey Nisnevich, who was motivated by the theory of adeles.

The most important of these invariants are homotopy groups, homology, and cohomology. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.

Examples of topologies include the Zariski topology in algebraic geometry that reflects the algebraic nature of varieties, and the topology on a differential manifold in differential topology where each point within the space is contained in an open set that is homeomorphic to an open ball in a finite-dimensional Euclidean space.

The integers with their usual topology are a discrete subgroup of the real numbers. In mathematics, a discrete subgroup of a topological group G is a subgroup H such that there is an open cover of G in which every open subset contains exactly one element of H ; in other words, the subspace topology of H in G is the discrete topology. For example, the integers, Z, form a discrete subgroup of the reals, R (with the standard metric topology), but the rational numbers, Q, do not. A discrete group is a topological group G equipped with the discrete topology.

Given an interior algebra we can form the Stone representation of its underlying Boolean algebra and then extend this to a topological field of sets by taking the topology generated by the complexes corresponding to the open elements of the interior algebra (which form a base for a topology). These complexes are then precisely the open complexes and the construction produces a Stone field representing the interior algebra - the Stone representation. (The topology of the Stone representation is also known as the McKinsey-Tarski Stone topology after the mathematicians who first generalized Stone's result for Boolean algebras to interior algebras and should not be confused with the Stone topology of the underlying Boolean algebra of the interior algebra which will be a finer topology).

For a given set A, whether a subset of Aω will be determined depends to some extent on its topological structure. For the purposes of Gale-Stewart games, the set A is endowed with the discrete topology, and Aω endowed with the resulting product topology, where Aω is viewed as a countably infinite topological product of A with itself. In particular, when A is the set {0,1}, the topology defined on Aω is exactly the ordinary topology on Cantor space, and when A is the set of natural numbers, it is the ordinary topology on Baire space. The set Aω can be viewed as the set of paths through a certain tree, which leads to a second characterization of its topology.

The pre-order can be obtained as the specialization pre-order of the McKinsey-Tarski topology. The Esakia duality can be recovered via a functor that replaces the field of sets with the Boolean space it generates. Via a functor that instead replaces the pre-order with its corresponding Alexandrov topology, an alternative representation of the interior algebra as a field of sets is obtained where the topology is the Alexandrov bico- reflection of the McKinsey-Tarski topology. The approach of formulating a topological duality for interior algebras using both the Stone topology of the Jónsson–Tarski approach and the Alexandrov topology of the pre-order to form a bi-topological space has been investigated by G. Bezhanishvili, R.Mines, and P.J. Morandi.

In 1975 he earned his Ph.D. from the University of California, Berkeley as a student of Robion Kirby. In topology, he has worked on handlebody theory, low-dimensional manifolds, symplectic topology, G2 manifolds. In the topology of real-algebraic sets, he and Henry C. King proved that every compact piecewise-linear manifold is a real-algebraic set; they discovered new topological invariants of real- algebraic sets.S. Akbulut and H.C. King, Topology of real algebraic sets, MSRI Publications, 25.

In mathematics, geometry and topology is an umbrella term for the historically distinct disciplines of geometry and topology, as general frameworks allow both disciplines to be manipulated uniformly, most visibly in local to global theorems in Riemannian geometry, and results like the Gauss–Bonnet theorem and Chern–Weil theory. Sharp distinctions between geometry and topology can be drawn, however, as discussed below. It is also the title of a journal Geometry & Topology that covers these topics.

In mathematics, and especially general topology, the prime integer topology and the relatively prime integer topology are examples of topologies on the set of positive whole numbers, i.e. the set }. To give the set Z+ a topology means to say which subsets of Z+ are "open", and to do so in a way that the following axioms are met: # The union of open sets is an open set. # The finite intersection of open sets is an open set.

In this case, we take to be the vector space instead of } so that the notation is unambiguous (whether denotes the space induced by a radial disk or the space induced by a bounded disk). The quotient topology on (inherited from 's original topology) is finer (in general, strictly finer) than the norm topology.

Going in the other direction, suppose (X, ≤) is a preordered set. Define a topology τ on X by taking the open sets to be the upper sets with respect to ≤. Then the relation ≤ will be the specialization preorder of (X, τ). The topology defined in this way is called the Alexandrov topology determined by ≤.

Topology and Its Applications is a peer-reviewed mathematics journal publishing research on topology. It was established in 1971 as General Topology and Its Applications, and renamed to its current title in 1980. The journal currently publishes 18 issues each year in one volume. It is indexed by Scopus, Mathematical Reviews, and Zentralblatt MATH.

If x is a point of a scheme X, then the local ring of x in the Nisnevich topology is the henselization of the local ring of x in the Zariski topology.

Fig 3: Topology of the ultra-sparse matrix. Characteristics of the Ultra Sparse Matrix Converter topology are 9 Transistors, 18 Diodes, and 7 Isolated Driver Potentials. The significant limitation of this converter topology compared to the Sparse Matrix Converter is the restriction of its maximal phase displacement between input voltage and input current which is restricted to ± 30°.

Twinax is a bus topology that requires termination to function properly. Most Twinax T-connectors have an automatic termination feature. For use in buildings wired with Category 3 or higher twisted pair there are baluns that convert Twinax to twisted pair and hubs that convert from a bus topology to a star topology. Twinax was designed by IBM.

Since compact sets in X are finite subsets, all compact subsets are closed, another condition usually related to Hausdorff separation axiom. The cocountable topology on a countable set is the discrete topology. The cocountable topology on an uncountable set is hyperconnected, thus connected, locally connected and pseudocompact, but neither weakly countably compact nor countably metacompact, hence not compact.

A fairly precise date can be supplied in the internal notes of the Bourbaki group. While topology was still combinatorial in 1942, it had become algebraic by 1944. gives documentation (translated into English from French originals). Azriel Rosenfeld (1973) proposed digital topology for a type of image processing that can be considered as a new development of combinatorial topology.

A topological space X is called orderable if there exists a total order on its elements such that the order topology induced by that order and the given topology on X coincide. The order topology makes X into a completely normal Hausdorff space. The standard topologies on R, Q, Z, and N are the order topologies.

Metric spaces embody a metric, a precise notion of distance between points. Every metric space can be given a metric topology, in which the basic open sets are open balls defined by the metric. This is the standard topology on any normed vector space. On a finite-dimensional vector space this topology is the same for all norms.

41 ;Discrete topology: See discrete space. ;Disjoint union topology: See Coproduct topology. ;Dispersion point: If X is a connected space with more than one point, then a point x of X is a dispersion point if the subspace X − {x} is hereditarily disconnected (its only connected components are the one- point sets). ;Distance: See metric space.

Lattice topology filters are not very common. The reason for this is that they require more components (especially inductors) than other designs. Ladder topology is much more popular. However, they do have the property of being intrinsically balanced and a balanced version of another topology, such as T-sections, may actually end up using more inductors.

"Late Maastrichtian pterosaurs from North Africa and mass extinction of Pterosauria at the Cretaceous-Paleogene boundary." PLoS Biology, 16(3): e2001663. Topology 1: Andres et al. (2014). Topology 2: Longrich et al. (2018).

In 1934, Krieger published an English translation of Sierpinski's book Introduction to General Topology. She also translated General Topology by Sierpinski in 1952, adding a 30-page appendix on infinite cardinals and ordinals.

Under the subspace topology, the singleton set {–1} is open in Y, but under the induced order topology, any open set containing –1 must contain all but finitely many members of the space.

This conflicts with the use of the term in topology.

They are used in algebraic number theory and algebraic topology.

This was dictated by the topology and the desired densities.

He is known for the Puppe sequence in algebraic topology.

Topology of beta-strands in "Greek-key" protein motif. Protein topology is a property of protein molecule that does not change under deformation (without cutting or breaking a bond). Two main topology frameworks have been developed and applied to protein molecules: 1) Knot theory which categorises chain entanglements 2) Circuit topology which categorises intra-chain contacts based on their arrangements. The usage of knot theory is however limited to a small percentage of proteins as most of them are unknot.

Any self-stabilizing algorithm recovers from a change in the network topology – the system configuration after a topology change can be treated just like any other arbitrary starting configuration. However, in a self- stabilizing algorithm, the convergence after a single change in the network topology may be as slow as the convergence from an arbitrary starting state. In the study of superstabilizing algorithms, special attention is paid to the time it takes to recover from a single change in the network topology.

In mathematics, in the realm of point-set topology, a Toronto space is a topological space that is homeomorphic to every proper subspace of the same cardinality. There are five homeomorphism classes of countable Toronto spaces, namely: the discrete topology, the indiscrete topology, the cofinite topology and the upper and lower topologies on the natural numbers. The only countable Hausdorff Toronto space is the discrete space.. The Toronto space problem asks for an uncountable Toronto Hausdorff space that is not discrete..

DNA topology is the tertiary conformations of DNA, such as supercoiling, knotting, and catenation. Topology of DNA can be disrupted by most metabolic processes: RNA polymerase can cause positive supercoils by over-winding the DNA in front of the enzyme, and can also cause negative supercoils by under-winding the DNA behind the enzyme. DNA polymerase has the same effect in DNA replication. Positive and negative supercoiling balance out the entire global topology of the DNA, so overall, the topology remains the same.

A hermitian matrix of unit determinant and having positive eigenvalues can be uniquely expressed as the exponential of a traceless hermitian matrix, and therefore the topology of this is that of -dimensional Euclidean space. Section 2.5 Since SU(n) is simply connected, Proposition 13.11 we conclude that is also simply connected, for all n. The topology of is the product of the topology of SO(n) and the topology of the group of symmetric matrices with positive eigenvalues and unit determinant. Since the latter matrices can be uniquely expressed as the exponential of symmetric traceless matrices, then this latter topology is that of -dimensional Euclidean space.

For any totally ordered set X we can define the open intervals (a, b) = {x : a < x and x < b}, (−∞, b) = {x : x < b}, (a, ∞) = {x : a < x} and (−∞, ∞) = X. We can use these open intervals to define a topology on any ordered set, the order topology. When more than one order is being used on a set one talks about the order topology induced by a particular order. For instance if N is the natural numbers, < is less than and > greater than we might refer to the order topology on N induced by < and the order topology on N induced by > (in this case they happen to be identical but will not in general). The order topology induced by a total order may be shown to be hereditarily normal.

Thomas "Tim" Daniel Cochran (April 7, 1955 – December 16, 2014) was a professor of Mathematics at Rice University specializing in topology, especially low-dimensional topology, the theory of knots and links and associated algebra.

He was an Invited Speaker of the International Congress of Mathematicians in 1974 in Vancouver. He is regarded as a founder of categorical topology, which deals with general topology using the methods of category theory.

All these topologies are identical. Series topology is a general name. Voltage divider or potential divider is used for circuits of that purpose. L-section is a common name for the topology in filter design.

The simplest option is to take the usual product topology. Another option is to take the topology generated by open sets consisting of functions whose value is specified on less than λ elements of λ.

There is a unique topology on the empty set ∅. The only open set is the empty one. Indeed, this is the only subset of ∅. Likewise, there is a unique topology on a singleton set {a}.

There are two main types of topology for a spacetime M.

This is a list of general topology topics, by Wikipedia page.

Ultrafilters have many applications in set theory, model theory, and topology.

Finite topology is a mathematical concept which has several different meanings.

A thickening of the trefoil knot Topology is the field concerned with the properties of continuous mappings, and can be considered a generalization of Euclidean geometry. In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness. The field of topology, which saw massive development in the 20th century, is in a technical sense a type of transformation geometry, in which transformations are homeomorphisms. This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'.

In mathematics, in the field of potential theory, the fine topology is a natural topology for setting the study of subharmonic functions. In the earliest studies of subharmonic functions, namely those for which \Delta u \ge 0, where \Delta is the Laplacian, only smooth functions were considered. In that case it was natural to consider only the Euclidean topology, but with the advent of upper semi-continuous subharmonic functions introduced by F. Riesz, the fine topology became the more natural tool in many situations.

Milnor was awarded the 2011 Abel Prize, for his "pioneering discoveries in topology, geometry and algebra." Reacting to the award, Milnor told the New Scientist "It feels very good," adding that "[o]ne is always surprised by a call at 6 o'clock in the morning." In 2013 he became a fellow of the American Mathematical Society, for "contributions to differential topology, geometric topology, algebraic topology, algebra, and dynamical systems".2014 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2013-11-04.

The XntAp is equally similar to the SLC8 and SLC24 protein families by amino acid sequence, but the predicted TM topology is more like that of SLC24, but the similarity is at best weak and the relationship is very distant. The AtMHX protein from plants also shares a distant relationship with the SLC8 proteins. The TM topology of the XNTA proteinThe figure shows the predicted TM topology of XntAp. Adapted from Haynes et al. (2002), this figure shows the computer predicted membrane topology of XntAp in Paramecium.

Every contractible space is simply connected. ;Coproduct topology: If {Xi} is a collection of spaces and X is the (set-theoretic) disjoint union of {Xi}, then the coproduct topology (or disjoint union topology, topological sum of the Xi) on X is the finest topology for which all the injection maps are continuous. ;Cosmic space: A continuous image of some separable metric space. ;Countable chain condition: A space X satisfies the countable chain condition if every family of non-empty, pairswise disjoint open sets is countable.

Spacetime topology is the topological structure of spacetime, a topic studied primarily in general relativity. This physical theory models gravitation as the curvature of a four dimensional Lorentzian manifold (a spacetime) and the concepts of topology thus become important in analysing local as well as global aspects of spacetime. The study of spacetime topology is especially important in physical cosmology.

If I is an ideal in a commutative ring R, the powers of I form topological neighborhoods of 0 which allow R to be viewed as a topological ring. This topology is called the I-adic topology. R can then be completed with respect to this topology. Formally, the I-adic completion is the inverse limit of the rings R/In.

If a hierarchy topology is used then updates would flow in a tree like structure through specific paths. In a flat topology it is entirely a matter of the peer relationships between nodes as to how updates take place. In a hybrid topology consisting of both flat and hierarchy topologies updates may take place through specific paths and between peers.

That is, every element of τ1 is also an element of τ2. Then the topology τ1 is said to be a coarser (weaker or smaller) topology than τ2, and τ2 is said to be a finer (stronger or larger) topology than τ1. There are some authors, especially analysts, who use the terms weak and strong with opposite meaning (Munkres, p. 78).

Ethernet Ring Protection Switching, or ERPS, is an effort at ITU-T under G.8032 Recommendation to provide sub-50ms protection and recovery switching for Ethernet traffic in a ring topology and at the same time ensuring that there are no loops formed at the Ethernet layer. G.8032v1 supported a single ring topology and G.8032v2 supports multiple rings/ladder topology.

A topological space X is said to be locally regular if and only if each point, x, of X has a neighbourhood that is regular under the subspace topology. Equivalently, a space X is locally regular if and only if the collection of all open sets that are regular under the subspace topology forms a base for the topology on X.

J. 10 (1943), 761–785. See Marcel Erné, Closure, in Frédéric Mynard, Elliott Pearl (Editors), Beyond Topology, Contemporary mathematics vol. 486, American Mathematical Society, 2009, p.170ff With the advancement of categorical topology in the 1980s, Alexandrov spaces were rediscovered when the concept of finite generation was applied to general topology and the name finitely generated spaces was adopted for them.

A set is totally bounded if, given any positive radius, it is covered by finitely many balls of that radius. The open balls of a metric space can serve as a base, giving this space a topology, the open sets of which are all possible unions of open balls. This topology on a metric space is called the topology induced by the metric .

The term topology also refers to a specific mathematical idea central to the area of mathematics called topology. Informally, a topology tells how elements of a set relate spatially to each other. The same set can have different topologies. For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set with different topologies.

In 2013, a topology by Andres & Myers placed Uktenadactylus within the family Ornithocheiridae, as the sister taxon of Coloborhynchus clavirostris, though in the analysis, Uktenadactylus was indentified as Coloborhynchus wadleighi. In 2019 however, a different topology by Jacobs et al. also recovered Uktenadactylus within the Ornithocheiridae, but as the sister taxon of several Coloborhynchus species. Topology 1: Andres & Myers (2013).

Algebraic & Geometric Topology is a peer-reviewed mathematics journal published quarterly by Mathematical Sciences Publishers. Established in 2001, the journal publishes articles on topology. Its 2018 MCQ was 0.82, and its 2018 impact factor was 0.709.

The metric space (RX, d) is complete. The ring RX is compact if and only if R is finite. This follows from Tychonoff's theorem and the characterisation of the topology on RX as a product topology.

In mathematics, the Tate topology is a Grothendieck topology of the space of maximal ideals of a k-affinoid algebra, whose open sets are the admissible open subsets and whose coverings are the admissible open coverings.

Membrane topology prediction for FAM155B. In terms of membrane topology, the N- and C-termini appear to be located extracellularly while the protein sequence between the transmembrane domains appears to be located in the cytoplasmic region.

Lately, topology control algorithms have been divided into two subproblems: topology construction, in charge of the initial reduction, and topology maintenance, in charge of the maintenance of the reduced topology so that characteristics like connectivity and coverage are preserved. This is the first stage of a topology control protocol. Once the initial topology is deployed, specially when the location of the nodes is random, the administrator has no control over the design of the network; for example, some areas may be very dense, showing a high number of redundant nodes, which will increase the number of message collisions and will provide several copies of the same information from similarly located nodes. However, the administrator has control over some parameters of the network: transmission power of the nodes, state of the nodes (active or sleeping), role of the nodes (Clusterhead, gateway, regular), etc.

The group of units R× of R is a topological group when endowed with the topology coming from the embedding of R× into the product R × R as (x,x−1). However, if the unit group is endowed with the subspace topology as a subspace of R, it may not be a topological group, because inversion on R× need not be continuous with respect to the subspace topology. An example of this situation is the adele ring of a global field; its unit group, called the idele group, is not a topological group in the subspace topology. If inversion on R× is continuous in the subspace topology of R then these two topologies on R× are the same.

Here, the basic open sets are the half open intervals [a, b). This topology on R is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it. If Γ is an ordinal number, then the set Γ = [0, Γ) may be endowed with the order topology generated by the intervals (a, b), [0, b) and (a, Γ) where a and b are elements of Γ. Outer space of a free group Fn consists of the so-called "marked metric graph structures" of volume 1 on Fn.

Morgan's best-known work deals with the topology of complex manifolds and algebraic varieties. In the 1970s, Dennis Sullivan developed the notion of a minimal model of a differential graded algebra.Dennis Sullivan. Infinitesimal computations in topology. Inst.

This is not, however, possible in this case where the Y-Δ transform is needed in addition to the series and parallel rules.Farago, pp.18–21 Redifon, p.22 The Y topology is also called star topology.

Jean Cerf (born in 1928) is a French mathematician, specializing in topology.

The NUMAlink 3 system interconnect uses a fat tree hypercube network topology.

This allows a relation between such morphisms and covering maps in topology.

The following phylogenetic analysis follows the topology of Andres et al. (2014).

The distillation regions and the nodes are the topology of the mixture.

The Mackey topology has an application in economies with infinitely many commodities.

Her research in categorical topology was published in 1986 by Horst Herrlich.

James Raymond Munkres (born August 18, 1930) is a Professor Emeritus of mathematics at MITJames Munkres, MIT Mathematics, mit.edu and the author of several texts in the area of topology, including Topology (an undergraduate- level text), Analysis on Manifolds, Elements of Algebraic Topology, and Elementary Differential Topology. He is also the author of Elementary Linear Algebra. Munkres completed his undergraduate education at Nebraska Wesleyan UniversityMathematics, The Tech, Volume 119, Issue 33, August 27, 1999 and received his Ph.D. from the University of Michigan in 1956; his advisor was Edwin E. Moise.

If is a pseudometric on a set then collection of open balls: :}, as ranges over and ranges over the positive real numbers, forms a basis for a topology on that is called the -topology or the pseudometric topology on induced by . :Convention: If is a pseudometric space and is treated as a topological space, then unless indicated otherwise, it should be assumed that is endowed with the topology induced by . ;Pseudometrizable space :Definition: A topological space is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp.

In chemistry, topology provides a way of describing and predicting the molecular structure within the constraints of three-dimensional (3-D) space. Given the determinants of chemical bonding and the chemical properties of the atoms, topology provides a model for explaining how the atoms ethereal wave functions must fit together. Molecular topology is a part of mathematical chemistry dealing with the algebraic description of chemical compounds so allowing a unique and easy characterization of them. Topology is insensitive to the details of a scalar field, and can often be determined using simplified calculations.

In other words, one terminal has been split into two terminals and the network has effectively been converted to a 4-terminal network. This topology is known as unbalanced topology and is opposed to balanced topology. Balanced topology requires, referring to Figure 3, that the impedance measured between terminals 1 and 3 is equal to the impedance measured between 2 and 4. This is the pairs of terminals not forming ports: the case where the pairs of terminals forming ports have equal impedance is referred to as symmetrical.

Lattice filter topology A lattice phase equaliser or lattice filter is an example of an all-pass filter. That is, the attenuation of the filter is constant at all frequencies but the relative phase between input and output varies with frequency. The lattice filter topology has the particular property of being a constant-resistance network and for this reason is often used in combination with other constant resistance filters such as bridge-T equalisers. The topology of a lattice filter, also called an X-section is identical to bridge topology.

However, many arguments in algebraic geometry work better for projective varieties, essentially because projective varieties are compact. From the 1920s to the 1940s, B. L. van der Waerden, André Weil and Oscar Zariski applied commutative algebra as a new foundation for algebraic geometry in the richer setting of projective (or quasi-projective) varieties.Dieudonné (1985), section VII.4. In particular, the Zariski topology is a useful topology on a variety over any algebraically closed field, replacing to some extent the classical topology on a complex variety (based on the topology of the complex numbers).

If is a TVS whose bounded subsets are exactly the same as its weakly bounded subsets (e.g. if is a Hausdorff locally convex space), then a -topology on (as defined in this article) is a polar topology and conversely, every polar topology if a -topology. Consequently, in this case the results mentioned in this article can be applied to polar topologies. However, if is a TVS whose bounded subsets are not exactly the same as its weakly bounded subsets, then the notion of "bounded in " is stronger than the notion of "-bounded in " (i.e.

Many sets of linear operators in functional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function. Any local field has a topology native to it, and this can be extended to vector spaces over that field. Every manifold has a natural topology since it is locally Euclidean. Similarly, every simplex and every simplicial complex inherits a natural topology from Rn. The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic variety.

In mathematics, more specifically general topology, the nested interval topology is an example of a topology given to the open interval (0,1), i.e. the set of all real numbers x such that . The open interval (0,1) is the set of all real numbers between 0 and 1; but not including either 0 or 1. To give the set (0,1) a topology means to say which subsets of (0,1) are "open", and to do so in a way that the following axioms are met: # The union of open sets is an open set.

From the previous characteristics of MyriaNed it can be derived that it uses a true mesh topology. The advantage of such a topology is reliability, and coping with mobility, because of the redundant communication paths in the network.

In 2014 she became the inaugural winner of the Joan & Joseph Birman Research Prize in Topology and Geometry, given biennially by the Association for Women in Mathematics to an outstanding early-career female researcher in topology and geometry.

The study of homotopy groups of spheres builds on a great deal of background material, here briefly reviewed. Algebraic topology provides the larger context, itself built on topology and abstract algebra, with homotopy groups as a basic example.

In algebraic geometry, the h topology is a Grothendieck topology introduced by Vladimir Voevodsky to study the homology of schemes. It combines several good properties possessed by its related "sub"topologies, such as the qfh and cdh topologies.

Similarly, we have the dual definition of the weak topology on induced by (and ), which is denoted by \sigma(Y,X,b) or simply : it is the weakest TVS topology on making all maps continuous, as ranges over .

Let C be a category and let J be a Grothendieck topology on C. The pair (C, J) is called a site. A presheaf on a category is a contravariant functor from C to the category of all sets. Note that for this definition C is not required to have a topology. A sheaf on a site, however, should allow gluing, just like sheaves in classical topology.

Edelsbrunner has over 100 research publicationsDBLP: Herbert Edelsbrunner. and is an ISI highly cited researcher.ISI highly cited researcher: Herbert Edelsbrunner. He has also published four books on computational geometry: Algorithms in Combinatorial Geometry (Springer-Verlag, 1987, ), Geometry and Topology for Mesh Generation (Cambridge University Press, 2001, ), Computational Topology (American Mathematical Society, 2009, 978-0821849255) and A Short Course in Computational Geometry and Topology (Springer-Verlag, 2014, ).

A topology table is used by routers that route traffic in a network. It consists of all routing tables inside the Autonomous System where the router is positioned. Each router using the routing protocol EIGRP then maintains a topology table for each configured network protocol — all routes learned, that are leading to a destination are found in the topology table. EIGRP must have a reliable connection.

A three-dimensional depiction of a thickened trefoil knot, the simplest non- trivial knot. Knot theory is an important part of low-dimensional topology. In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups.

In 1996, his book "Inverse Spectra", was published. A powerful method used in many areas of topology, the notion of an inverse spectra was originally introduced by Lefschetz. Dr. Chigogidze was on the editorial board of the JP Journal of Geometry and Topology, and also on the Tbilisi Mathematical Journal. He had served as an Editor in Chief of "Topology and its Applications" since 2009.

The area of network tomography also includes that of inferring network topology using end-to-end probes. Topology discovery is a tradeoff between accuracy vs overhead. In network tomography, the emphasis is to achieve as accurate a picture of the network with minimal overhead. In comparison, other network topology discovery techniques using SNMP or Route analytics aim for greater accuracy with less emphasis on overhead reduction.

Physical topology is the placement of the various components of a network (e.g., device location and cable installation), while logical topology illustrates how data flows within a network. Distances between nodes, physical interconnections, transmission rates, or signal types may differ between two different networks, yet their logical topologies may be identical. A network’s physical topology is a particular concern of the physical layer of the OSI model.

Tree network topology A tree network, or star-bus network, is a hybrid network topology in which star networks are interconnected via bus networks. Tree networks are hierarchical, and each node can have an arbitrary number of child nodes.

Any such sequence belongs to the Hilbert space ℓ2, so the Hilbert cube inherits a metric from there. One can show that the topology induced by the metric is the same as the product topology in the above definition.

Their local objects are affine schemes or prime spectra which are locally ringed spaces which form a category which is antiequivalent to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field k, and the category of finitely generated reduced k-algebras. The gluing is along Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The Zariski topology in the set theoretic sense is then replaced by a Grothendieck topology. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely the étale topology, and the two flat Grothendieck topologies: fppf and fpqc; nowadays some other examples became prominent including Nisnevich topology.

In topology, a field within mathematics, desuspension is an operation inverse to suspension.

JTS has been ported to the .NET Framework as the Net Topology Suite.

Efstratia (Effie) Kalfagianni is a Greek American mathematician specializing in low-dimensional topology.

There are many deep connections with contact topology, which is the "opposite" concept.

The cladogram below is from the topology recovered by Kellner et al. (2019).

The cladogram below follows the topology from a 2011 analysis by Susumu Tomiya.

The size of metadata can also be decreased by restricting the communication topology; for instance, in a star, tree or linear topology, a single scalar suffices. The search for efficient implementations of causal consistency is a very active research area.

The product topology on a product of topological spaces \prod X_i has basis \prod U_i where U_i \subseteq X_i is open, and cofinitely many U_i = X_i. The analog (without requiring that cofinitely many are the whole space) is the box topology.

This ability, known as topology simplification, was first identified by Rybenkov et al.Rybenkov, V. V., Ullsperger, C., Vologodskii, A. V., & Cozzarelli, N. R. (1997). Simplification of DNA topology below equilibrium values by type II topoisomerases. Science, 277(5326), 690–693.

All these topologies can be viewed as a short section of a ladder topology. Longer sections would normally be described as ladder topology. These kinds of circuits are commonly analysed and characterised in terms of a two-port network.Farago, pp.

In the mathematical field of point-set topology, a continuum (plural: "continua") is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. Continuum theory is the branch of topology devoted to the study of continua.

Every complete lattice is also a bounded lattice, which is to say that it has a greatest and least element. In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology.

A Seifert surface bounded by a set of Borromean rings; these surfaces can be used as tools in geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.

His involvement with topology and the Poincaré conjecture led to the creation of the Whitehead manifold. The definition of crossed modules is due to him. He also made important contributions in differential topology, particularly on triangulations and their associated smooth structures.

IEEE Computer Graphics and Applications, 11(6), pp.44-51. including computer aided geometric design, topology-based shape matching,M. Hilaga, Y. Shinagawa, T. Kohmura and T.L. Kunii, 2001, August. Topology matching for fully automatic similarity estimation of 3D shapes.

Accessed November 27, 2008 Since 1986 Vogtmann has been a co-organizer of the annual conference called the Cornell Topology FestivalCornell Topology Festival, grant summary. Cornell University. Accessed November 28, 2008 that usually takes places at Cornell University each May.

In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non- Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff.

Mesh network topology. Red nodes denote corners, blue border and gray interior. The mesh is another popular form of network topology, especially in parallel systems, redundant memory systems and interconnection networks.N. Santoro, Design and Analysis of Distributed Algorithms, Wiley, 2006.

For example, "pointless topology" (in other words, point-free topology, or locale theory) starts with a single base set whose elements imitate open sets in a topological space (but are not sets of points); see also mereotopology and point-free geometry.

The basis sets in the product topology have almost the same definition as the above, except with the qualification that all but finitely many Ui are equal to the component space Xi. The product topology satisfies a very desirable property for maps fi : Y → Xi into the component spaces: the product map f: Y → X defined by the component functions fi is continuous if and only if all the fi are continuous. As shown above, this does not always hold in the box topology. This actually makes the box topology very useful for providing counterexamples--many qualities such as compactness, connectedness, metrizability, etc., if possessed by the factor spaces, are not in general preserved in the product with this topology.

In 2008-09, concerned about the rapidly growing demand for cloud computing, the IT industry was contemplating converting the prevalent multi-rooted tree based network topology, of oversubscribed data center networks, to a fat tree topology or to a non-blocking full bisection bandwidth clos topology. The difficulty was that re-hauling these networks with complex aggregate layer network switches, as required in a fat tree network topology, or with thousands of simpler commodity network switches inter-connected with miles of wires as required in a clos network topology, was very expensive. In a study of a production mega data center, published in Oct. 2009, Bahl, Kandula, and Padhye showed that barring a few outliers, traffic demands could be met in existing, slightly-oversubscribed data center networks.

However, this name is frequently changed according to the types of sets that make up (e.g. the "topology of uniform convergence on compact sets" or the "topology of compact convergence", see the footnote for more detailsIn practice, usually consists of a collection of sets with certain properties and this name is changed appropriately to reflect this set so that if, for instance, is the collection of compact subsets of (and is a topological space), then this topology is called the topology of uniform convergence on the compact subsets of .). A subset of is said to be fundamental with respect to if each is a subset of some element in . In this case, the collection can be replaced by without changing the topology on .

In mathematics, a quasi-topology on a set X is a function that associates to every compact Hausdorff space C a collection of mappings from C to X satisfying certain natural conditions. A set with a quasi-topology is called a quasitopological space. They were introduced by Spanier, who showed that there is a natural quasi-topology on the space of continuous maps from one space to another.

A topological group, , is a topological space that is also a group such that the group operation (in this case product): :, and inversion map: :, are continuousi.e. Continuous means that for any open set , is open in the domain of . Here is viewed as a topological space with the product topology. Such a topology is said to be compatible with the group operations and is called a group topology.

Also, any set can be given the trivial topology (also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence and net in this topology converges to every point of the space. This example shows that in general topological spaces, limits of sequences need not be unique. However, often topological spaces must be Hausdorff spaces where limit points are unique.

Ladder topology, often called Cauer topology after Wilhelm Cauer (inventor of the elliptic filter), was in fact first used by George Campbell (inventor of the constant k filter). Campbell published in 1922 but had clearly been using the topology for some time before this. Cauer first picked up on ladders (published 1926) inspired by the work of Foster (1924). There are two forms of basic ladder topologies; unbalanced and balanced.

In algebraic topology, several types of products are defined on homological and cohomological theories.

Shelly L. Harvey. On the cut number of a 3-manifold. Geometry & Topology, vol.

Lamination (topology), a partition of a closed subset of the surface into smooth curves.

Topics covered include analytic geometry, set theory, abstract algebra, group theory, topology, and probability.

The result relied heavily on techniques developed for studying the topology of Heegaard splittings.

Matroids have found applications in geometry, topology, combinatorial optimization, network theory and coding theory.

In 2006, the nine editorial board members of Oxford University's Elsevier-published mathematics journal Topology resigned because they agreed among themselves that Elsevier's publishing policies had "a significant and damaging effect on Topology reputation in the mathematical research community." An Elsevier spokesperson disputed this, saying that "this still constitutes a pretty rare occurrence" and that the journal "is actually available today to more people than ever before". Journalists recognize this event as part of the precedent to The Cost of Knowledge campaign. In 2008, the Journal of Topology started independently of Elsevier, and Topology ended publication in 2009.

Switch box topology Whenever a vertical and a horizontal channel intersect, there is a switch box. In this architecture, when a wire enters a switch box, there are three programmable switches that allow it to connect to three other wires in adjacent channel segments. The pattern, or topology, of switches used in this architecture is the planar or domain-based switch box topology. In this switch box topology, a wire in track number one connects only to wires in track number one in adjacent channel segments, wires in track number 2 connect only to other wires in track number 2 and so on.

In algebra, the following construction is common: one starts with a commutative ring R containing an ideal I, and then considers the I-adic topology on R: a subset U of R is open if and only if for every x in U there exists a natural number n such that x + In ⊆ U. This turns R into a topological ring. The I-adic topology is Hausdorff if and only if the intersection of all powers of I is the zero ideal (0). The p-adic topology on the integers is an example of an I-adic topology (with I = (p)).

Traditionally a neural network topology is chosen by a human experimenter, and effective connection weight values are learned through a training procedure. This yields a situation whereby a trial and error process may be necessary in order to determine an appropriate topology. NEAT is an example of a topology and weight evolving artificial neural network (TWEANN) which attempts to simultaneously learn weight values and an appropriate topology for a neural network. In order to encode the network into a phenotype for the GA, NEAT uses a direct encoding scheme which means every connection and neuron is explicitly represented.

Möbius strips, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing. A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology.

The logical view is that OSPF creates something of a spider web or star topology of many areas all attached directly to Area Zero and IS-IS, by contrast, creates a logical topology of a backbone of Level 2 routers with branches of Level 1–2 and Level 1 routers forming the individual areas. IS-IS also differs from OSPF in the methods by which it reliably floods topology and topology change information through the network. However, the basic concepts are similar. OSPF has a larger set of extensions and optional features specified in the protocol standards.

An embedded Lie subgroup is closed Corollary 15.30. so a subgroup is an embedded Lie subgroup if and only if it is closed. Equivalently, is an embedded Lie subgroup if and only if its group topology equals its relative topology. Problem 2.

The second analysis placed Ferrodraco as a basal member of the Anhangueria, and sister taxon to the polytomy that comprises Anhanguera, Coloborhynchus and Ornithocheirus. Topology 1: First analysis by Pentland et al. (2019). Topology 2: Second analysis by Pentland et al. (2019).

Robert F. Riley (December 22, 1935-March 4, 2000) was an American mathematician. He is known for his work in low-dimensional topology using computational tools and hyperbolic geometry, being one of the inspirations for William Thurston's later breakthroughs in 3-dimensional topology.

Applied topology explains how large molecules reach their final shapes and how biological molecules achieve their activity. Circuit topology is a topological property of folded linear polymers. This notion has been applied to structural analysis of biomolecules such as proteins and RNAs.

The Sierpiński space S is a special case of both the finite particular point topology (with particular point 1) and the finite excluded point topology (with excluded point 0). Therefore, S has many properties in common with one or both of these families.

This was the first resolution of one of the Hilbert Problems. Dehn's interests later turned to topology and combinatorial group theory. In 1907 he wrote with Poul Heegaard the first book on the foundations of combinatorial topology, then known as analysis situs.

Yuli B. Rudyak is a professor of Mathematics at the University of Florida in Gainesville, Florida. He obtained his doctorate from Moscow State University under the supervision of M. M. Postnikov. His main research interests are geometry and topology and symplectic topology.

This topology can only be done in an IC, as the matching has to be extremely close and cannot be achieved with discretes. Another topology is the Wilson current mirror. The Wilson mirror solves the Early effect voltage problem in this design.

The results of Cannon's paper were used by Cannon, Bryant and Lacher to prove (1979)J. W. Cannon, J. L. Bryant and R. C. Lacher, The structure of generalized manifolds having nonmanifold set of trivial dimension. Geometric topology (Proc. Georgia Topology Conf.

This is typically the case for the Zariski topology on the spectrum of a ring. For the usual basis of this topology, every finite intersection of basis elements is a basis element. Therefore bases are sometimes required to be stable by finite intersection.

Dieter Puppe (middle) with John Frank Adams in 1962, in Aarhus. Siegmund Dieter Puppe (16 December 1930 – 13 August 2005Mathematik in der Heidelberger Akademie der Wissenschaften. 2014, pg. 60) was a German mathematician who worked in algebraic topology, differential topology and homological algebra.

A quotient space is defined as follows: if X is a topological space and Y is a set, and if f : X→ Y is a surjective function, then the quotient topology on Y is the collection of subsets of Y that have open inverse images under f. In other words, the quotient topology is the finest topology on Y for which f is continuous. A common example of a quotient topology is when an equivalence relation is defined on the topological space X. The map f is then the natural projection onto the set of equivalence classes. The Vietoris topology on the set of all non-empty subsets of a topological space X, named for Leopold Vietoris, is generated by the following basis: for every n-tuple U1, ..., Un of open sets in X, we construct a basis set consisting of all subsets of the union of the Ui that have non-empty intersections with each Ui. The Fell topology on the set of all non-empty closed subsets of a locally compact Polish space X is a variant of the Vietoris topology, and is named after mathematician James Fell.

Douglas Conner Ravenel (born 1947) is an American mathematician known for work in algebraic topology.

Alexander horned sphere The Alexander horned sphere is a pathological object in topology discovered by .

Advantages to this topology include tighter synchronization, as well as automatic infrastructure redundancy (see below).

In general several different uniform structures can be compatible with a given topology on X.

Eduard J. Zehnder is a Swiss mathematician, considered one of the founders of symplectic topology.

Any open subset of an n-manifold is an n-manifold with the subspace topology.

The fine topology was introduced in 1940 by Henri Cartan to aid in the study of thin sets and was initially considered to be somewhat pathological due to the absence of a number of properties such as local compactness which are so frequently useful in analysis. Subsequent work has shown that the lack of such properties is to a certain extent compensated for by the presence of other slightly less strong properties such as the quasi-Lindelöf property. In one dimension, that is, on the real line, the fine topology coincides with the usual topology since in that case the subharmonic functions are precisely the convex functions which are already continuous in the usual (Euclidean) topology. Thus, the fine topology is of most interest in \R^n where n\geq 2.

In the Zariski topology on the affine plane, this graph of a polynomial is closed. In algebraic geometry and commutative algebra, the Zariski topology is a topology on algebraic varieties, introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space, called the spectrum of the ring. The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces.

Suppose that is a topological vector space and is a convex balanced and radial set. Then } is a neighborhood basis at the origin for some locally convex topology on . This TVS topology is given by the Minkowski functional formed by , , which is a seminorm on defined by . The topology is Hausdorff if and only if is a norm, or equivalently, if and only if } or equivalently, for which it suffices that be bounded in .

The new combinatorial topology formally treated topological classes as abelian groups. Homology groups are finitely generated abelian groups, and homology classes are elements of these groups. The Betti numbers of the manifold are the rank of the free part of the homology group, and the non-orientable cycles are described by the torsion part. The subsequent spread of homology groups brought a change of terminology and viewpoint from "combinatorial topology" to "algebraic topology".

Since every vector topology is translation invariant (i.e. for all , the map defined by is a homeomorphism), to define a vector topology it suffices to define a neighborhood basis (or subbasis) for it at the origin. Note that in general, the set of all balanced and absorbing subsets of a vector space does not satisfy the conditions of this theorem and does not form a neighborhood basis at the origin for any vector topology.

Hess has worked and written extensively on topics in algebraic topology including homotopy theory, model categories and algebraic K-theory. She has also used the methods of algebraic topology and category theory to investigate homotopical generalizations of descent theory and Hopf–Galois extensions. In particular, she has studied generalizations of these structures for ring spectra and differential graded algebras. She has more recently used algebraic topology to understand structures in neurology and materials science.

In fact they are a base for the standard topology on the real numbers. However, a base is not unique. Many different bases, even of different sizes, may generate the same topology. For example, the open intervals with rational endpoints are also a base for the standard real topology, as are the open intervals with irrational endpoints, but these two sets are completely disjoint and both properly contained in the base of all open intervals.

Eduard Čech at the First International Topology conference in Moscow, 1935 Eduard Čech (; 29 June 1893 – 15 March 1960) was a Czech mathematician born in Stračov (then Bohemia, Austria-Hungary, now Czech Republic). His research interests included projective differential geometry and topology. He is especially known for the technique known as Stone–Čech compactification (in topology) and the notion of Čech cohomology. He was the first to publish a proof of Tychonoff's theorem in 1937.

Similarly, a space is R0 if and only if the specialization preorder is an equivalence relation. Given any equivalence relation on a finite set X the associated topology is the partition topology on X. The equivalence classes will be the classes of topologically indistinguishable points. Since the partition topology is pseudometrizable, a finite space is R0 if and only if it is completely regular. Non-discrete finite spaces can also be normal.

For proteins with disulfide bonds, these bonds could be considered as contacts. In a context where beta-beta interactions in proteins are more relevant, these interactions are used to define the circuit topology. As such, circuit topology framework can be applied to a wide range of applications including protein folding and analysis of genome architecture. In particular, data from Hi-C and related technologies can be readily analysed using circuit topology framework.

In particular, there must exist a nonzero functional on — that is, the continuous dual space is non-trivial. Considering with the weak topology induced by , then becomes locally convex; by the second bullet of geometric Hahn-Banach, the weak topology on this new space separates points. Thus with this weak topology becomes Hausdorff. This sometimes allows some results from locally convex topological vector spaces to be applied to non-Hausdorff and non-locally convex spaces.

Physically, AFDX can be a cascaded star topology of multiple dual redundant Ethernet switches; however, the AFDX Virtual links are modeled as time-switched single-transmitter bus connections, thus following the safety model of a single-transmitter bus topology previously used in aircraft. Logical topologies are often closely associated with media access control methods and protocols. Some networks are able to dynamically change their logical topology through configuration changes to their routers and switches.

All finite fields are locally compact since they can be equipped with the discrete topology. In particular, any field with the discrete topology is locally compact since every point is the neighborhood of itself, and also the closure of the neighborhood, hence is compact.

Most of Floyd's research is in the areas of geometric topology and geometric group theory. Floyd and Allen Hatcher classified all the incompressible surfaces in punctured-torus bundles over the circle.Floyd, W.; Hatcher, A. Incompressible surfaces in punctured-torus bundles. Topology and its Applications, vol.

In these models, there are non-zero transition amplitudes between two different topologies, or in other words, the topology changes. This and other similar results lead physicists to believe that any consistent quantum theory of gravity should include topology change as a dynamical process.

L-RNA is much more stable against degradation by RNase. Like other structured biopolymers such as proteins, one can define topology of a folded RNA molecule. This is often done based on arrangement of intra-chain contacts within a folded RNA, termed as circuit topology.

The geometry and topology of three-manifolds is a set of widely circulated but unpublished notes by William Thurston from 1978 to 1980 describing his work on 3-manifolds. The notes introduced several new ideas into geometric topology, including orbifolds, pleated manifolds, and train tracks.

To use Parasolid effectively, users need to have fundamental knowledge of CAD, computational geometry and topology.

Depending on the network topology (fully connected, hypercube, ring), different all-to-all algorithms are required.

Networks have several properties, including: number of nodes (oscillators), network topology, and coupling strength between oscillators.

This structure has a very complex topology composed of four beta-sheets and three alpha helices.

In mathematics, particularly in algebraic topology, Alexander–Spanier cohomology is a cohomology theory for topological spaces.

The McKinsey-Tarski topology of an interior algebra is the intersection of the former two topologies.

These functions, their negativity and minima have a direct interpretation in algebraic topology (Baudot & Bennequin, 2015).

Every topological space is a dense subset of itself. For a set X equipped with the discrete topology, the whole space is the only dense subset. Every non- empty subset of a set X equipped with the trivial topology is dense, and every topology for which every non-empty subset is dense must be trivial. Denseness is transitive: Given three subsets A, B and C of a topological space X with such that A is dense in B and B is dense in C (in the respective subspace topology) then A is also dense in C. The image of a dense subset under a surjective continuous function is again dense.

By definition, it is the coarsest topology on for which all maps in are continuous. That the vector space operations are continuous in this topology follows from properties 2 and 3 above. It can easily be seen that the resulting topological vector space is "locally convex" in the sense of the first definition given above because each is absolutely convex and absorbent (and because the latter properties are preserved by translations). Note that it is possible for a locally convex topology on a space to be induced by a family of norms but for to not be normable (that is, to have its topology be induced by a single norm).

In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber)Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag (See Chapter 11 for construction.) is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces . It is dual to the mapping cone. In particular, given such a map, define the mapping path space to be the set of pairs where and is a path such that . We give a topology by giving it the subspace topology as a subset of (where is the space of paths in which as a function space has the compact-open topology).

The cocountable topology or countable complement topology on any set X consists of the empty set and all cocountable subsets of X, that is all sets whose complement in X is countable. It follows that the only closed subsets are X and the countable subsets of X. Every set X with the cocountable topology is Lindelöf, since every nonempty open set omits only countably many points of X. It is also T1, as all singletons are closed. If X is an uncountable set, any two open sets intersect, hence the space is not Hausdorff. However, in the cocountable topology all convergent sequences are eventually constant, so limits are unique.

In a co-located topology, you use a percentage of available memory on existing web or worker roles for Caching. The following diagram shows Caching in a co-located topology. The cloud service has two roles: Web1 and Worker1. There are two running instances of each role.

In topology, a branch of mathematics, a topological monoid is a monoid object in the category of topological spaces. In other words, it is a monoid with a topology with respect to which the monoid's binary operation is continuous. Every topological group is a topological monoid.

This is because despite it making addition and negation continuous, a group topology on a vector space may fail to make scalar multiplication continuous. For instance, the discrete topology on any non- trivial vector space makes addition and negation continuous but do not make scalar multiplication continuous.

If is a topological vector space (TVS) (where note in particular that is assumed to be a vector topology) then the following are equivalent: 1. is pseudometrizable (i.e. the vector topology is induced by a pseudometric on ). 2. has a countable neighborhood base at the origin. 3.

There are several commercial topology optimization software on the market. Most of them use topology optimization as a hint how the optimal design should look like, and manual geometry re-construction is required. There are a few solutions which produce optimal designs ready for Additive Manufacturing.

Mary Ellen Rudin (December 7, 1924 – March 18, 2013) was an American mathematician known for her work in set-theoretic topology. In 2013, Elsevier established the Mary Ellen Rudin Young Researcher Award which is awarded annually to a young researcher, mainly in fields adjacent to general topology.

In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C that makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site. Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology.

Almost all spaces encountered in analysis are Hausdorff; most importantly, the real numbers (under the standard metric topology on real numbers) are a Hausdorff space. More generally, all metric spaces are Hausdorff. In fact, many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdorff condition explicitly stated in their definitions. A simple example of a topology that is T1 but is not Hausdorff is the cofinite topology defined on an infinite set.

This also includes the collection and provision of Ethernet port- related statistics for network maintenance. With these mechanisms, the topology of a Profinet IO network can be read out at any time and the status of the individual connections can be monitored. If the network topology is known, automatic addressing of the nodes can be activated by their position in the topology. This considerably simplifies device replacement during maintenance, since no more settings need to be made.

In this method, computational methods are used for topology optimization of the structure. Expected loading and desired motion and force transmission is inputted and the system is optimized for minimum stresses, weight, and accuracy. More advanced methods first optimize the underlying linkage configuration and then optimize the topology around that configuration. Other optimization techniques focus topology optimization of the flexure joints by taking as input a rigid mechanism and replacing all the rigid joints with optimized flexure joints.

Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds. More specifically, differential topology considers the properties and structures that require only a smooth structure on a manifold to be defined. Smooth manifolds are "softer" than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology.

Filters designed using network synthesis usually repeat the simplest form of L-section topology though component values may change in each section. Image designed filters, on the other hand, keep the same basic component values from section to section though the topology may vary and tend to make use of more complex sections. L-sections are never symmetrical but two L-sections back-to-back form a symmetrical topology and many other sections are symmetrical in form.

In the 1960s, Grothendieck defined the notion of a site, meaning a category equipped with a Grothendieck topology. A site C axiomatizes the notion of a set of morphisms Vα → U in C being a covering of U. A topological space X determines a site in a natural way: the category C has objects the open subsets of X, with morphisms being inclusions, and with a set of morphisms Vα → U being called a covering of U if and only if U is the union of the open subsets Vα. The motivating example of a Grothendieck topology beyond that case was the étale topology on schemes. Since then, many other Grothendieck topologies have been used in algebraic geometry: the fpqc topology, the Nisnevich topology, and so on. The definition of a sheaf works on any site.

It utilises the classic SSL G Series center compressor design elements within a Super-Analogue design topology.

Ribbon theory is a strand of mathematics within topology that has seen particular application as regards DNA.

Blender has multi-res digital sculpting, which includes dynamic topology, maps baking, remeshing, re-symmetrize, and decimation.

It has a simple topology consisting of three α-helices that form a well-packed hydrophobic core.

The Fréchet filter is named after the French mathematician Maurice Fréchet (1878-1973), who worked in topology.

Michael Jerome Hopkins (born April 18, 1958) is an American mathematician known for work in algebraic topology.

A notion of a p-space has been introduced by Alexander Arhangelskii.Encyclopedia of General Topology, p. 278.

For example, any complex algebraic variety X, with its classical (Euclidean) topology, is compactifiable in this sense.

CAA replaces the function provided by Topology Services (topsvcs) in RSCT in previous releases of PowerHA/HACMP .

Tobias Holck Colding (born 1963) is a Danish mathematician working on geometric analysis, and low-dimensional topology.

Lanthanosuchoidea is a node-based taxon defined in 1997 as "the most recent common ancestor of Lanthanosuchus, Lanthaniscus, and Acleistorhinus". The cladogram below follows the topology from a 2011 analysis by Ruta et al. The cladogram below follows the topology from a 2016 analysis by MacDougall et al.

However, it should be noticed, some manufacturing constraints such as minimal feature size also need to be considered during the topology optimization process. Since the topology optimization can help designers to get an optimal complex geometry for additive manufacturing, this technique can be considered one of DfAM methods.

Grothendieck & Raynaud, SGA 1, Exposé XII. (The first group here is defined using the Zariski topology, and the second using the classical (Euclidean) topology.) For example, the equivalence between algebraic and analytic coherent sheaves on projective space implies Chow's theorem that every closed analytic subspace of CPn is algebraic.

Loanwords can severely affect the topology of a tree so efforts are made to exclude borrowings. However, undetected ones sometimes still exist. McMahon and McMahon Language Classification by Numbers showed that around 5% borrowing can affect the topology while 10% has significant effects. In networks borrowing produces reticulations.

The federation can be loosely integrated between institutions, tightly integrated or a combination of both. Monadic topology has a central repository that all collected data is fed into. The central repository then responds to all queries for data. There are no replicas in this topology as compared to others.

In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra.

Available on-line at: Mocavo.com hence founding the field of algebraic topology. In 1916 Oswald Veblen applied the algebraic topology of Poincaré to Kirchhoff's analysis.Oswald Veblen, The Cambridge Colloquium 1916, (New York : American Mathematical Society, 1918-1922), vol 5, pt. 2 : Analysis Situs, "Matrices of orientation", pp. 25-27.

In mathematics, particularly topology, a topological space X is locally normal if intuitively it looks locally like a normal space. More precisely, a locally normal space satisfies the property that each point of the space belongs to a neighbourhood of the space that is normal under the subspace topology.

Differential topology is the study of global geometric invariants without a metric or symplectic form. Differential topology starts from the natural operations such as Lie derivative of natural vector bundles and de Rham differential of forms. Beside Lie algebroids, also Courant algebroids start playing a more important role.

A phylogenetic analysis by Pentland et al. in 2019 for example, had found Cearadactylus in a derived position within the Anhangueria, just outside the Ornithocheirae, which is the clade that contains the families Ornithocheiridae and Anhangueridae. Topology 1: Pereda-Suberbiola et al. (2012). Topology 2: Pentland et al. (2019).

Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces.Allen Hatcher, Algebraic topology. (2002) Cambridge University Press, xii+544 pp. . The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

Within the scheme, modules can be filled in with more or less complex algorithms which control the process. The run time engine loads the modules. The user can adapt the modules to the topology. For complex topology, multiple modules can be used or parallel loops can be implemented.

In particular, when is a topological space and is the set of all compact subsets (resp. finite subsets) of then this characterizes the compact-open topology (resp. topology of pointwise convergence) on . Stepping away from the function spaces topologies defined above, suppose that is a metric space with metric .

Paul Alexander Schweitzer, S. J., (born 21 July 1937, Yonkers, New York) is an American mathematician, specializing in differential topology, geometric topology, and algebraic topology. He has done research on foliations, knot theory, and 3-manifolds. In 1974 he found a counterexample to the Seifert conjecture that every non-vanishing vector field on the 3-sphere has a closed integral curve. In 1995 he demonstrated that Sergei Novikov's compact leaf theorem cannot be generalized to manifolds with dimension greater than 3\.

Ole Sigmund (born May 28, 1966) is a Danish Professor in Mechanical Engineering who has made fundamental contributions to the field of Topology optimization, including microstructure design, nano optics, photonic crystals, Matlab codes, acoustics, and fluids. In 2003 he co-authored the highly cited book "Topology Optimization - Theory, Methods and Applications" with Martin P. Bendsøe. His research group was the first to achieve giga-resolution topology optimization, making it for the first time possible to optimize an entire Boing 777 wing structure.

The Journal of Topology is a peer-reviewed scientific journal which publishes papers of high quality and significance in topology geometry and adjacent areas of mathematics. It was established in 2008, when the Editorial Board of Topology resigned due to the increasing costs of Elsevier's subscriptions. The journal is owned and managed by the London Mathematical Society and produced, distributed, sold and marketed by John Wiley & Sons. It appears quarterly with articles published individually online prior to appearing in a printed issue.

In topology, the cartesian product of topological spaces can be given several different topologies. One of the more obvious choices is the box topology, where a base is given by the Cartesian products of open sets in the component spaces.Willard, 8.2 pp. 52-53, Another possibility is the product topology, where a base is given by the Cartesian products of open sets in the component spaces, only finitely many of which can be not equal to the entire component space.

The Origin 2000's network topology is a bristled fat hypercube. In configurations with more than 64 processors, a hierarchical fat hypercube network topology is used instead. Additional NUMAlink cables, called Xpress links can be installed between unused Standard Router ports to reduce latency and increase bandwidth. Xpress links can only be used in systems that have 16 or 32 processors, as these are the only configurations with a network topology that enables unused ports to be used in such a way.

A Storm application is designed as a "topology" in the shape of a directed acyclic graph (DAG) with spouts and bolts acting as the graph vertices. Edges on the graph are named streams and direct data from one node to another. Together, the topology acts as a data transformation pipeline. At a superficial level the general topology structure is similar to a MapReduce job, with the main difference being that data is processed in real time as opposed to in individual batches.

An important example of a non-normal topology is given by the Zariski topology on an algebraic variety or on the spectrum of a ring, which is used in algebraic geometry. A non-normal space of some relevance to analysis is the topological vector space of all functions from the real line R to itself, with the topology of pointwise convergence. More generally, a theorem of Arthur Harold Stone states that the product of uncountably many non-compact metric spaces is never normal.

A topological field of sets is called algebraic if and only if there is a base for its topology consisting of complexes. If a topological field of sets is both compact and algebraic then its topology is compact and its compact open sets are precisely the open complexes. Moreover, the open complexes form a base for the topology. Topological fields of sets that are separative, compact and algebraic are called Stone fields and provide a generalization of the Stone representation of Boolean algebras.

In mathematics, a Zariski geometry consists of an abstract structure introduced by Ehud Hrushovski and Boris Zilber, in order to give a characterisation of the Zariski topology on an algebraic curve, and all its powers. The Zariski topology on a product of algebraic varieties is very rarely the product topology, but richer in closed sets defined by equations that mix two sets of variables. The result described gives that a very definite meaning, applying to projective curves and compact Riemann surfaces in particular.

This board then launched the new Journal of Topology at a far lower price, under the auspices of the London Mathematical Society.Journal of Topology (pub. London Mathematical Society) After this mass resignation, Topology remained in circulation under a new editorial board until 2009, when the last issue was published. The elevated pricing of field journals in economics, most of which are published by Elsevier, was one of the motivations that moved the American Economic Association to launch the American Economic Journal in 2009.

There are many ways of defining a topology on R, the set of real numbers. The standard topology on R is generated by the open intervals. The set of all open intervals forms a base or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set.

In functional analysis and related areas of mathematics a FK-space or Fréchet coordinate space is a sequence space equipped with a topological structure such that it becomes a Fréchet space. FK-spaces with a normable topology are called BK-spaces. There exists only one topology to turn a sequence space into a Fréchet space, namely the topology of pointwise convergence. Thus the name coordinate space because a sequence in an FK-space converges if and only if it converges for each coordinate.

Neighbor joining may be viewed as a greedy heuristic for the Balanced Minimum Evolution (BME) criterion. For each topology, BME defines the tree length (sum of branch lengths) to be a particular weighted sum of the distances in the distance matrix, with the weights depending on the topology. The BME optimal topology is the one which minimizes this tree length. Neighbor joining at each step greedily joins that pair of taxa which will give the greatest decrease in the estimated tree length.

Specializing in geometry, topology and group theory, Bridson is best known for his work in Geometric Group Theory.

Topology soon became a separate field of major importance, rather than a sub-field of geometry or analysis.

The study and generalization of this formula, specifically by Cauchy and L'Huilier, is at the origin of topology.

The modelled ERK3/MAPK6 kinase domain is predicted to fold with a topology similar to other MAP kinases.

Alexandroff plank in topology, an area of mathematics, is a topological space that serves as an instructive example.

Dolecki, An Initiation into Convergence Theory, in F. Mynard, E. Pearl (editors), Beyond Topology, AMS, Contemporary Mathematics, 2009. .

In topology, the Bing metrization theorem, named after R. H. Bing, characterizes when a topological space is metrizable.

It can be noted that the entire network topology is almost regular with an O(log n) diameter.

The sets Sk(U) form a basis for the Whitney Ck- topology on C∞(M,N)., p. 42.

In a subsequent paperMladen Bestvina and Michael Handel. Train-tracks for surface homeomorphisms. Topology, vol. 34 (1995), no.

With superstabilizing systems, there is a passage predicate that is always satisfied while the system's topology is reconfigured.

The excluded point topology on any finite set is a completely normal T0 space which is non-discrete.

He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to topology".

Fomenko is an accomplished painter and illustrator whose work often depicts objects from mathematics, many related to topology.

Let C be any category. The Yoneda embedding gives a functor Hom(−, X) for each object X of C. The canonical topology is the biggest (finest) topology such that every representable presheaf, i.e. presheaf of the form Hom(−, X), is a sheaf. A covering sieve or covering family for this site is said to be strictly universally epimorphic because it consists of the legs of a colimit cone (under the full diagram on the domains of its constituent morphisms) and these colimits are stable under pullbacks along morphisms in C. A topology that is less fine than the canonical topology, that is, for which every covering sieve is strictly universally epimorphic, is called subcanonical.

In mathematics, a topological space X is uniformizable if there exists a uniform structure on X that induces the topology of X. Equivalently, X is uniformizable if and only if it is homeomorphic to a uniform space (equipped with the topology induced by the uniform structure). Any (pseudo)metrizable space is uniformizable since the (pseudo)metric uniformity induces the (pseudo)metric topology. The converse fails: There are uniformizable spaces that are not (pseudo)metrizable. However, it is true that the topology of a uniformizable space can always be induced by a family of pseudometrics; indeed, this is because any uniformity on a set X can be defined by a family of pseudometrics.

Every uniform space X becomes a topological space by defining a subset O of X to be open if and only if for every x in O there exists an entourage V such that V[x] is a subset of O. In this topology, the neighbourhood filter of a point x is {V[x] : V ∈ Φ}. This can be proved with a recursive use of the existence of a "half-size" entourage. Compared to a general topological space the existence of the uniform structure makes possible the comparison of sizes of neighbourhoods: V[x] and V[y] are considered to be of the "same size". The topology defined by a uniform structure is said to be induced by the uniformity. A uniform structure on a topological space is compatible with the topology if the topology defined by the uniform structure coincides with the original topology.

Mary-Elizabeth Hamstrom (May 24, 1927 – December 2, 2009) was an American mathematician known for her contributions to topology, and particularly to point-set topology and the theory of homeomorphism groups of manifolds. She was for many years a professor of mathematics at the University of Illinois at Urbana–Champaign.

Bezhanishvili et al. (2010). If is a Priestley space, then both and are spectral spaces. Conversely, given a spectral space , let denote the patch topology on ; that is, the topology generated by the subbasis consisting of compact open subsets of and their complements. Let also denote the specialization order of .

In mathematics, more specifically in topology, the equivariant stable homotopy theory is a subfield of equivariant topology that studies a spectrum with group action instead of a space with group action, as in stable homotopy theory. The field has become more active recently because of its connection to algebraic K-theory.

In geometric topology, however, holes are determined differently. In this field, the genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non- intersecting closed simple curves without rendering the resultant manifold disconnected.Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.

In mathematics, particularly topology, a topological space X is locally regular if intuitively it looks locally like a regular space. More precisely, a locally regular space satisfies the property that each point of the space belongs to an open subset of the space that is regular under the subspace topology.

The bridged T topology is used for delay equalisation, particularly the differential delay between two landlines being used for stereophonic sound broadcasts. This application requires that the filter has a linear phase response with frequency (i.e., constant group delay) over a wide bandwidth and is the reason for choosing this topology.

Type I KH domain (βααββα topology) and type II KH domain (αββααβ topology). For both classes, the GXXG loop, the flanking helices, the β-strand and the variable loop between β2 and β3 (type I) or between α2 and β2 (type II) play a very important role in recognizing RNA.

Their phylogenetic tree, based on a dataset derived from the separate analyses of Jalil (1997), David Dilkes (1998), and Michael Benton & Jackie Allen (1997), is reproduced below, at left. Topology A: Rieppel et al. (2008) Topology B: Liu et al. (2017) Liu and colleagues conducted a separate phylogenetic analysis in 2017.

This implies e. g. that every completely metrizable topological vector space is complete. Indeed, a topological vector space is called complete iff its uniformity (induced by its topology and addition operation) is complete; the uniformity induced by a translation-invariant metric that induces the topology coincides with the original uniformity.

Centralized Management of Bandwidth Resources and Uniform Service Provisioning In the IP hard pipe solution, the NMS centrally manages bandwidth resources and uniformly provisions services. Provisioning hard pipe service involves two steps: 1\. Establish a hard pipe plane. In the physical network topology, establish the hard pipe topology as designed. 2\.

If is endowed with this topology, then a subset is open in this topology if and only if for every , there exists some such that . Moreover, for any , . On a uniform space every convergent filter is a Cauchy filter. Moreover, every cluster point of a Cauchy filter is a limit point.

A tree topology (a.k.a. hierarchical topology) can be viewed as a collection of star networks arranged in a hierarchy. This tree has individual peripheral nodes (e.g. leaves) which are required to transmit to and receive from one other node only and are not required to act as repeaters or regenerators.

Mark Edward Mahowald (December 1, 1931 – July 20, 2013) was an American mathematician known for work in algebraic topology.

A Hausdorff TVS is metrizable if and only if its topology can be induced by a single topological string.

These proteins have a structural motif consisting of a 2-layer sandwich structure with an alpha/beta plait topology.

Laura Christine Kinsey is an American mathematician specializing in topology. She is a professor of mathematics at Canisius College.

From 1948 to 1980 he was the head of the topology section. One of his students was Andrzej Mostowski.

Finally, detecting nontrivial patterns of reciprocity can reveal possible mechanisms and organizing principles that shape the observed network's topology.

Francis Bonahon, Oberwolfach 1987 Francis Bonahon (9 September 1955, Tarbes) is a French mathematician, specializing in low-dimensional topology.

Topology control has to be executed periodically in order to preserve the desired properties such as connectivity, coverage, density.

"The Topology of Anne Wilson's Topologies." Anne Wilson: Unfoldings Ed. Lisa Tung. Boston:: Mass Art, 2002. pp. 36-41.

In the mathematical field of topology, there are various notions of a P-space and of a p-space.

Lyudmila Keldysh (aka Ljudmila Vsevolodovna Keldyš; ) (1904-1976) was a Russian mathematician known for set theory and geometric topology.

In mathematics, weak convergence in a Hilbert space is convergence of a sequence of points in the weak topology.

There are a number of basic definitions of K-theory: two coming from topology and two from algebraic geometry.

However, a dcpo with the Scott topology is a Hausdorff space if and only if the order is trivial. The Scott-open sets form a complete lattice when ordered by inclusion. For any Kolmogorov space, the topology induces an order relation on that space, the specialization order: if and only if every open neighbourhood of x is also an open neighbourhood of y. The order relation of a dcpo D can be reconstructed from the Scott-open sets as the specialization order induced by the Scott topology.

In 2001 he was awarded the Oswald Veblen Prize in Geometry from the AMS for his work in symplectic and contact topology. In particular for his proof of the symplectic rigidity and the development of 3-dimensional contact topology. In 2009 he received the Doctorat Honoris Causa from the ENS Lyon and in 2017 the Doctorat Honoris Causa from the University of Uppsala. In 2013 Eliashberg shared with Helmut Hofer the Heinz Hopf Prize from the ETH, Zurich, for their pioneering research in symplectic topology.

In 1924 he was awarded the Bôcher Memorial Prize for his work in mathematical analysis. The Lefschetz fixed point theorem, now a basic result of topology, was developed by him in papers from 1923 to 1927, initially for manifolds. Later, with the rise of cohomology theory in the 1930s, he contributed to the intersection number approach (that is, in cohomological terms, the ring structure) via the cup product and duality on manifolds. His work on topology was summed up in his monograph Algebraic Topology (1942).

In 1996, dissatisfied with the rapidly rising fees charged by the major publishers of mathematical research journals, Rourke decided to start his own journal, and was ably assisted by Robion Kirby, John Jones and Brian Sanderson. That journal became Geometry & Topology. Under Rourke's leadership, GT has become a leading journal in its field while remaining one of the least expensive per page. GT was joined in 1998 by a proceedings and monographs series, Geometry & Topology Monographs, and in 2000 by a sister journal, Algebraic & Geometric Topology.

Saying that Ψ maps from V to , or in other words, that Ψ(x) is continuous on for every , is a reasonable minimal requirement on the topology of , namely that the evaluation mappings : \varphi \in V' \mapsto \varphi(x), \quad x \in V , be continuous for the chosen topology on . Further, there is still a choice of a topology on , and continuity of Ψ depends upon this choice. As a consequence, defining reflexivity in this framework is more involved than in the normed case.

Even though linear continua are important in the study of ordered sets, they do have applications in the mathematical field of topology. In fact, we will prove that an ordered set in the order topology is connected if and only if it is a linear continuum (notice the 'if and only if' part). We will prove one implication, and leave the other one as an exercise. (Munkres explains the second part of the proof ) Theorem Let X be an ordered set in the order topology.

Liu graduated from Peking University in 1963, majoring in mathematics. He was assigned to Sichuan University after his graduation. His research field was topology and fuzzy mathematics, mainly in the algebra problem of unclear topology, embedded theory and nonclear convex sets. Liu was the deputy president of Sichuan University between 1989 and 2005.

In the dedicated topology, you define a worker role that is dedicated to Caching. This means that all of the worker role's available memory is used for the Caching and operating overhead. The following diagram shows Caching in a dedicated topology. The cloud service shown has three roles: Web1, Worker1, and Cache1.

Thus the dreams of the early topologists have long been regarded as a mirage. Cubical higher homotopy groupoids are constructed for filtered spaces in the book Nonabelian algebraic topology cited below, which develops basic algebraic topology, including higher analogues to the Seifert–van Kampen theorem, without using singular homology or simplicial approximation.

During normal operation, the network works in the Ring-Closed status (Figure 1). In this status, one of the MRM ring ports is blocked, while the other is forwarding. Conversely, both ring ports of all MRCs are forwarding. Loops are avoided because the physical ring topology is reduced to a logical stub topology.

From 1970 to 1975 he was a professor at Saarland University. In 1975 he became a professor at the University of Göttingen. Tammo tom Dieck is a world-class expert in algebraic topology and author of several widely-used textbooks in topology. He has done research on Lie groups, G-structures, and cobordism.

5–6 Figure 1.1. T,Y and Star topologies are all identical. This example also demonstrates a common convention of naming topologies after a letter of the alphabet to which they have a resemblance. Greek alphabet letters can also be used in this way, for example Π (pi) topology and Δ (delta) topology.

The side of an anti-prism forms a topology which, in this sense, is an anti-ladder. Anti- ladder topology finds an application in voltage multiplier circuits, in particular the Cockcroft-Walton generator. There is also a full-wave version of the Cockcroft-Walton generator which uses a double anti-ladder topology.Campbell, pp.

For example, the lower limit topology on the real line is first-countable, separable, and Lindelöf, but not second-countable. For metric spaces, however, the properties of being second-countable, separable, and Lindelöf are all equivalent.Willard, theorem 16.11, p. 112 Therefore, the lower limit topology on the real line is not metrizable.

For instance force/velocity is mechanical impedance. The mobility analogy does not preserve this analogy between impedances across domains, but it does have another advantage over the impedance analogy. In the mobility analogy the topology of networks is preserved, a mechanical network diagram has the same topology as its analogous electrical network diagram.

Throughout this section we will assume that and are topological vector spaces. will be a non-empty collection of subsets of directed by inclusion. :Notation: will denote the vector space of all continuous linear maps from into . If is given the -topology inherited from then this space with this topology is denoted by .

The order topology and metric topology on are the same. As a topological space, the real line is homeomorphic to the open interval . The real line is trivially a topological manifold of dimension . Up to homeomorphism, it is one of only two different connected 1-manifolds without boundary, the other being the circle.

There are several slightly different sorts of collapsing algebras. If κ and λ are cardinals, then the Boolean algebra of regular open sets of the product space κλ is a collapsing algebra. Here κ and λ are both given the discrete topology. There are several different options for the topology of κλ.

Four examples and two non- examples of topologies on the three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing.

The research of Wu includes the following fields: algebraic topology, algebraic geometry, game theory, history of mathematics, automated theorem proving. His most important contributions are to algebraic topology. The Wu class and the Wu formula are named after him. In the field of automated theorem proving, he is known for Wu's method.

The set (0,1) and the empty set ∅ are required to be open sets, and so we define (0,1) and ∅ to be open sets in this topology. The other open sets in this topology are all of the form where n is a positive whole number greater than or equal to two i.e. .

William Schumacher Massey (August 23, 1920 \- June 17, 2017) was an American mathematician, known for his work in algebraic topology. The Massey product is named for him. He worked also on the formulation of spectral sequences by means of exact couples, and wrote several textbooks, including A Basic Course in Algebraic Topology ().

A distributed star is a network topology that is composed of individual networks that are based upon the physical star topology connected in a linear fashion – i.e., 'daisy-chained' – with no central or top level connection point (e.g., two or more 'stacked' hubs, along with their associated star connected nodes or 'spokes').

Skeleton of Cryptoclidus oxoniensis (AMNH 995) Life restoration The cladogram below follows the topology from Benson et al. (2012) analysis.

Topology-based methods broadly make the assumption that nodes with similar network structure are more likely to form a link.

Paul Olum (August 16, 1918 – January 19, 2001) was an American mathematician (algebraic topology), professor of mathematics, and university administrator.

Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander.

Ramaswamy S. Vaidyanathaswamy (1894–1960) was an Indian mathematician who wrote the first textbook of point-set topology in India.

The qfh topology is associated to families as above, with the further restriction that each p_i must be quasi-finite.

World Scientific. fuzzy topology,Chang, C.L. (1968) "Fuzzy topological spaces", J. Math. Anal. Appl., 24, 182—190.Liu, Y.-M.

The hypersimplices were first studied and named in the computation of characteristic classes (an important topic in algebraic topology), by ...

Georges de Rham (; 10 September 1903 – 9 October 1990) was a Swiss mathematician, known for his contributions to differential topology.

In mathematics, especially in topology, equidimensionality is a property of a space that the local dimension is the same everywhere.

T. Asselmeyer-Maluga and C. Brans, Exotic Smoothness and Physics : Differential Topology and Spacetime Models, World Scientific Press, Singapore(2007).

DDR2 Command/Address (CA) bank Fly-by topology similar to that of the DDR3 Command/Address (CA) bank The noise levels on a trace/network is highly dependent on the routing topology selected. In a point-to-point topology, the signal is routed from the transmitter directly to the receiver (this is applied in PCIe, RapidIO, GbE, DDR2/DDR3/DDR4 DQ/DQS etc.). A point-to-point topology has the least SI- problems since there is no large impedance matches being introduced by line T's (a two-way split of a trace). For interfaces where multiple packages are receiving from the same line, (for example with a backplane configuration), the line must be split at some point to service all receivers.

Multiprotocol BGP allows information about the topology of IP multicast-capable routers to be exchanged separately from the topology of normal IPv4 unicast routers. Thus, it allows a multicast routing topology different from the unicast routing topology. Although MBGP enables the exchange of inter-domain multicast routing information, other protocols such as the Protocol Independent Multicast family are needed to build trees and forward multicast traffic. Multiprotocol BGP is also widely deployed in case of MPLS L3 VPN, to exchange VPN labels learned for the routes from the customer sites over the MPLS network, in order to distinguish between different customer sites when the traffic from the other customer sites comes to the Provider Edge router (PE router) for routing.

In it meant what is now called a Grothendieck pretopology, and some authors still use this old meaning. modified the definition to use sieves rather than covers. Much of the time this does not make much difference, as each Grothendieck pretopology determines a unique Grothendieck topology, though quite different pretopologies can give the same topology.

A tethered topology would primarily be used for reachback and forwarding between the airborne fighter platform and supporting elements. A flat ad hoc topology would be used between airborne fighter platforms in a strike package or CAP for the more frequent information exchanges. The figure outlines the minimum equipment requirements to support a fighter platform.

An important method for calculating the various groups is the concept of stable algebraic topology, which finds properties that are independent of the dimensions. Typically these only hold for larger dimensions. The first such result was Hans Freudenthal's suspension theorem, published in 1937. Stable algebraic topology flourished between 1945 and 1966 with many important results .

Anderson took numerous courses outside his major, including courses in sociology, economics, and business, and he returned to the USC engineering program to take a course in mathematical topology for non-mathematicians. In this last course, Anderson would have encountered the Möbius strip, since it is a fundamental concept in the study of topology.

In mathematics, a compactly generated (topological) group is a topological group G which is algebraically generated by one of its compact subsets.. This should not be confused with the unrelated notion (widely used in algebraic topology) of a compactly generated space -- one whose topology is generated (in a suitable sense) by its compact subspaces.

Metric spaces are first countable since one can use balls with rational radius as a neighborhood base. The metric topology on a metric space M is the coarsest topology on M relative to which the metric d is a continuous map from the product of M with itself to the non-negative real numbers.

The current proliferation of 3D printer technology has allowed designers and engineers to use topology optimization techniques when designing new products. Topology optimization combined with 3D printing can result in lightweighting, improved structural performance and shortened design-to-manufacturing cycle. As the designs, while efficient, might not be realisable with more traditional manufacturing techniques.

A physical topology that contains switching or bridge loops is attractive for redundancy reasons, yet a switched network must not have loops. The solution is to allow physical loops, but create a loop-free logical topology using the shortest path bridging (SPB) protocol or the older spanning tree protocols (STP) on the network switches.

If is a topology on , then the pair is called a topological space. The notation may be used to denote a set endowed with the particular topology . The members of are called open sets in . A subset of is said to be closed if its complement is in (that is, its complement is open).

Many neuroevolution algorithms have been defined. One common distinction is between algorithms that evolve only the strength of the connection weights for a fixed network topology (sometimes called conventional neuroevolution), as opposed to those that evolve both the topology of the network and its weights (called TWEANNs, for Topology and Weight Evolving Artificial Neural Network algorithms). A separate distinction can be made between methods that evolve the structure of ANNs in parallel to its parameters (those applying standard evolutionary algorithms) and those that develop them separately (through memetic algorithms).

Layout of a grid low-voltage network A grid network is a computer network consisting of a number of computer systems connected in a grid topology. In a regular grid topology, each node in the network is connected with two neighbors along one or more dimensions. If the network is one-dimensional, and the chain of nodes is connected to form a circular loop, the resulting topology is known as a ring. Network systems such as FDDI use two counter- rotating token-passing rings to achieve high reliability and performance.

In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. Roughly speaking, two or more spaces may be considered together, each looking as it would alone. The name coproduct originates from the fact that the disjoint union is the categorical dual of the product space construction.

Dimca competed in the International Mathematical Olympiad in 1970, 1971, and 1972, earning two bronze medals and one silver medal.. He obtained his PhD in 1981 from the University of Bucharest; his thesis "Stable mappings and singularities", was written under the direction of Gheorghe Galbură. His Google Scholar h-index is 24. Dimca is a distinguished mathematician in algebra, geometry and topology.Bridging Algebra, Geometry, and Topology He has written three important books in this field: Sheaves in Topology, Singularities and Topology of Hypersurfaces and Topics on real and complex singularities.

It also indicates a connection between the combinatorics of J-holomorphic curves in the blow up of the projective plane and the numbers that appear as indices in embedded contact homology. With Katrin Wehrheim, she has challenged the foundational rigor of a classic proof in symplectic geometry.. With Dietmar Salamon, she co-authored two textbooks Introduction to Symplectic TopologyIntroduction to Symplectic Topology, 2nd edition, Oxford U. Press and J-Holomorphic Curves and Symplectic Topology. J-holomorphic curves and symplectic topology is a greatly expanded rewriting of the 1994 book J-holomorphic curves and quantum cohomology.

Beginning with the Jones polynomial, infinitely many new invariants of knots, links, and 3-manifolds were found during the 1980s. The study of these new `quantum' invariants expanded rapidly into a sub-discipline of low-dimensional topology called quantum topology. A quantum invariant is typically constructed from two ingredients: a formal sum of Jacobi diagrams (which carry a Lie algebra structure), and a representation of a ribbon Hopf algebra such as a quantum group. It is not clear a-priori why either of these ingredients should have anything to do with low-dimensional topology.

Fig 1: Schematic of the split-pi converter topology In electronics, a split-pi topology is a pattern of component interconnections used in a kind of power converter that can theoretically produce an arbitrary output voltage, either higher or lower than the input voltage. In practice the upper voltage output is limited to the voltage rating of components used. It is essentially a boost (step-up) converter followed by a buck (step-down) converter. The topology and use of MOSFETs make it inherently bi-directional which lends itself to applications requiring regenerative braking.

Cobordism is a much coarser equivalence relation than diffeomorphism or homeomorphism of manifolds, and is significantly easier to study and compute. It is not possible to classify manifolds up to diffeomorphism or homeomorphism in dimensions ≥ 4 – because the word problem for groups cannot be solved – but it is possible to classify manifolds up to cobordism. Cobordisms are central objects of study in geometric topology and algebraic topology. In geometric topology, cobordisms are intimately connected with Morse theory, and h-cobordisms are fundamental in the study of high-dimensional manifolds, namely surgery theory.

At the beginning of the 20th century, Henri Poincaré was working on the foundations of topology—what would later be called combinatorial topology and then algebraic topology. He was particularly interested in what topological properties characterized a sphere. Poincaré claimed in 1900 that homology, a tool he had devised based on prior work by Enrico Betti, was sufficient to tell if a 3-manifold was a 3-sphere. However, in a 1904 paper he described a counterexample to this claim, a space now called the Poincaré homology sphere.

Fat tree DCN architecture reduces the oversubscription and cross section bandwidth problem faced by the legacy three-tier DCN architecture. Fat tree DCN employs commodity network switches based architecture using Clos topology. The network elements in fat tree topology also follows hierarchical organization of network switches in access, aggregate, and core layers. However, the number of network switches is much larger than the three-tier DCN. The architecture is composed of k pods, where each pod contains, (k/2)2 servers, k/2 access layer switches, and k/2 aggregate layer switches in the topology.

Completeness is a property of the metric and not of the topology, meaning that a complete metric space can be homeomorphic to a non-complete one. An example is given by the real numbers, which are complete but homeomorphic to the open interval , which is not complete. In topology one considers completely metrizable spaces, spaces for which there exists at least one complete metric inducing the given topology. Completely metrizable spaces can be characterized as those spaces that can be written as an intersection of countably many open subsets of some complete metric space.

Importantly, the subspace topology C^k(K;U) inherits from C^k(U) (when it is viewed as a subset of C^k(U)) is identical to the subspace topology that it inherits from C^k(V) (when C^k(K;U) is viewed instead as a subset of C^k(V) via the identification). Thus the topology on C^k(K;U) is independent of the open subset of \R^n that contains . This justifies our practice of using C^k(K) instead of C^k(K;U).

In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology.

An elementary filter topology introduces a capacitor into the feedback path of an op-amp to achieve an unbalanced active implementation of a low-pass transfer function Electronic filter topology defines electronic filter circuits without taking note of the values of the components used but only the manner in which those components are connected. Filter design characterises filter circuits primarily by their transfer function rather than their topology. Transfer functions may be linear or nonlinear. Common types of linear filter transfer function are; high-pass, low-pass, bandpass, band- reject or notch and all-pass.

Fig. 11: Relations between mathematical spaces: locales, topoi etc In Grothendieck's work on the Weil conjectures, he introduced a new type of topology now called a Grothendieck topology. A topological space (in the ordinary sense) axiomatizes the notion of "nearness," making two points be nearby if and only if they lie in many of the same open sets. By contrast, a Grothendieck topology axiomatizes the notion of "covering". A covering of a space is a collection of subspaces that jointly contain all the information of the ambient space.

The SPB Link Metric sub-TLV occurs within the Multi-Topology Intermediate System Neighbor TLV or within the Extended IS Reachability TLV. Where this sub-TLV is not present for an IS-IS adjacency, then that adjacency will not carry SPB traffic for the given topology instance. There are multiple ECT algorithms defined for SPB; however, for the future, additional algorithms may be defined; similarly the SPB Adjacency Opaque Equal Cost Tree Algorithm TLV also occurs within the Multi-Topology Intermediate System TLV or the Extended IS Reachability TLV.

In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology. Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography.

Its topology is a poset topology or Alexandrov topology. An abstract cell complex is a particular case of a locally finite space in which the dimension is defined for each point. It was demonstrated that the dimension of a cell c of an abstract cell complex is equal to the length (number of cells minus 1) of the maximum bounding path leading from any cell of the complex to the cell c. The bounding path is a sequence of cells in which each cell bounds the next one.

Network topology is the arrangement of the elements (links, nodes, etc.) of a communication network. Network topology can be used to define or describe the arrangement of various types of telecommunication networks, including command and control radio networks, industrial fieldbusses and computer networks. Network topology is the topological structure of a network and may be depicted physically or logically. It is an application of graph theory wherein communicating devices are modeled as nodes and the connections between the devices are modeled as links or lines between the nodes.

Bernardo Uribe Jongbloed (born 1975) is a Colombian mathematician. Uribe's research deals with algebraic geometry and topology with string theory applications.

This space is complete, but not normable: indeed, every neighborhood of 0 in the product topology contains lines, i.e., sets for .

In topology, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres.

Włodzimierz Kuperberg (born January 19, 1941) is a professor of mathematics at Auburn University, with research interests in geometry and topology.

In: Topology. v. 21, 1982, pp. 235–243; Greg Kuperberg: Quadrisecants of knots and links. In: J. Knot Theory Ramifications. v.

In addition, roGFPs are used to investigate the topology of ER proteins, or to analyze the ROS production capacity of chemicals.

The successor points and zero are the isolated points of the class of ordinal numbers, with respect to the order topology..

Kazimierz Zarankiewicz in Moscow 1935 Kazimierz Zarankiewicz (2 May 1902 – 5 September 1959) was a Polish mathematician, interested primarily in topology.

In 1913, he published a seminal work in the field of topology of surface entitled On Cutting the Plane by Continua.

In the above example, one can take a nonzero from the kernel of . Consequently, the resulting topology need not be Hausdorff.

Charles Waldo Rezk (born 26 January 1969) is an American mathematician, specializing in algebraic topology, category theory, and spectral algebraic geometry.

Given an FK-space X and \omega with the topology of pointwise convergence the inclusion map :\iota:X \to \omega is continuous.

Andreas Thom is a German mathematician, working on geometric group theory, algebraic topology, ergodic theory of group actions, and operator algebras.

Arnold worked on dynamical systems theory, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics and singularity theory.

The program memsat_svm is used to predict the presence and topology of any transmembrane helices present in the user protein sequence.

Edwin Earl Floyd (8 May 1924, Eufaula, Alabama – 9 December 1990) was an American mathematician, specializing in topology (especially cobordism theory).

Alexander Nikolaevich Varchenko (, born February 6, 1949) is a Soviet and Russian mathematician working in geometry, topology, combinatorics and mathematical physics.

Brouwer fixed-point theorem is a fixed-point theorem in topology, named after Dutchman Luitzen Brouwer, who proved it in 1911.

The homotopy colimit of a sequence of spaces :X_1 \to X_2 \to \cdots, is the mapping telescope.Hatcher's Algebraic Topology, 4.G.

Serguei Barannikov (; born April 16, 1972) is a mathematician, known for his works in algebraic topology, algebraic geometry and mathematical physics.

The netkeeper is not involved in the processing of calls but stores the platform topology information, routing configuration and alarm information.

There are two running instances of each role. In this example, the cache is distributed across all instances of the dedicated Cache1 role. Represents a running Windows Azure cloud service that uses Caching with a dedicated topology. A dedicated topology has the advantage of scaling the caching tier independently of any other role in the cloud service.

Suppose that is a vector space over , where is either the real or complex numbers, and let (resp. ) denote the open (resp. closed) ball of radius in . A family of seminorms on vector space induces a canonical vector space topology on , called the initial topology induced by the seminorms, making it into a topological vector space (TVS).

Strictly finer than the manifold topology. It is therefore Hausdorff, separable but not locally compact. A base for the topology is sets of the form Y^+(p,U) \cup Y^-(p,U) \cup p for some point p \in M and some convex normal neighbourhood U \subset M. (Y^\pm denote the chronological past and future).

Andres & Myers (2013) presented a phylogenetic analysis that placed Pteranodontoidea within the group Pteranodontia, as the sister taxon of the family Nyctosauridae. In 2018 however, Longrich, Martill, and Andres revisited the classification, and concluded that Pteranodontoidea would be the more inclusive group containing both Ornithocheiromorpha and Pteranodontia. Topology 1: Andres & Myers (2013). Topology 2: Longrich, Martill, and Andres (2018).

Distributed firewalls have both strengths and weaknesses when compared to conventional firewalls. By far the biggest difference, of course, is their reliance on topology. If the network topology does not permit reliance on traditional firewall techniques, there is little choice. A more interesting question is how the two types compare in a closed, single-entry network.

There exist measurable spaces that are not Borel spaces, for any choice of topology on the underlying space.Jochen Wengenroth, Is every sigma-algebra the Borel algebra of a topology? Measurable spaces form a category in which the morphisms are measurable functions between measurable spaces. A function f:X \rightarrow Y is measurable if it pulls back measurable sets, i.e.

Characteristics of the Sparse Matrix Converter topology are 15 Transistors, 18 Diodes, and 7 Isolated Driver Potentials. Compared to the Direct matrix converter this topology provides identical functionality, but with a reduced number of power switches and the option of employing an improved zero DC-link current commutation scheme, which provides lower control complexity and higher safety and reliability.

Ecological topology is another theme of complex systems in biogeomorphology. This theme focuses on how the biota varies based on geographic location. This ecological topology is controlled by a concept called stability domain. Stability domain describes the interaction of a set species and certain abiotic factors that act as a medium to the function and structure of an environment.

For n leaves there are 1 • 3 • 5 • ... • (2n-3) different topologies. Enumerating them is not feasible already for a small number of leaves. Heuristic search methods are used to find a reasonably good topology. The evaluation of S for a given topology (which includes the computation of the branch lengths) is a linear least squares problem.

A 10 TMS model has been presented, but this model conflicts with the 14 TMS model proposed for AE1. The transmembrane topology of the human pancreatic electrogenic Na+:HO transporter, NBC1, has been studied. A TMS topology with N- and C-termini in the cytoplasm has been suggested. An extracellular loop determines the stoichiometry of Na+-HCO cotransporters.

Barry Charles Mazur (; born December 19, 1937) is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in number theory, Mazur's torsion theorem in arithmetic geometry, the Mazur swindle in geometric topology, and the Mazur manifold in differential topology.

In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties (e.g., it is neither Lindelöf nor separable). Therefore, it serves as one of the basic counterexamples of topology.

For every preordered set X = we always have W(T(X)) = X, i.e. the preorder of X is recovered from the topological space T(X) as the specialization preorder. Moreover for every Alexandrov-discrete space X, we have T(W(X)) = X, i.e. the Alexandrov topology of X is recovered as the topology induced by the specialization preorder.

When the set has some structure (such as a group operation or a topology) and the equivalence relation is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories.

The space }, along with the double origin topology is an example of a Hausdorff space, although it is not completely Hausdorff. In terms of compactness, the space }, along with the double origin topology fails to be either compact, paracompact or locally compact, however, X is second countable. Finally, it is an example of an arc connected space.

The set S is not closed in the euclidean topology since it does not contain the origin which is a limit point of S, but the set is closed in the fine topology in \R^n. In comparison, it is not possible in \R^2 to construct such a connected set which is thin at the origin.

Casson has worked in both high-dimensional manifold topology and 3- and 4-dimensional topology, using both geometric and algebraic techniques. Among other discoveries, he contributed to the disproof of the manifold Hauptvermutung, introduced the Casson invariant, a modern invariant for 3-manifolds, and Casson handles, used in Michael Freedman's proof of the 4-dimensional Poincaré conjecture.

Hybrid topology is also known as hybrid network.Hybrid networks combine two or more topologies in such a way that the resulting network does not exhibit one of the standard topologies (e.g., bus, star, ring, etc.). For example, a tree network (or star-bus network) is a hybrid topology in which star networks are interconnected via bus networks.

Józef Henryk Przytycki (; ; born 14 October 1953 in Warsaw, Poland), is a mathematician specializing in the fields of knot theory and topology.

Grothendieck also saw how to phrase the definition of covering abstractly; this is where the definition of a Grothendieck topology comes from.

Information about convergence of nets and filters, such as definitions and properties, can be found in the article about filters in topology.

With this information, a message can be routed to the destination without knowledge of the network topology or a prior route discovery.

In algebraic topology, a branch of mathematics, a connective spectrum is a spectrum whose homotopy sets \pi_k of negative degrees are zero..

In mathematics, more specifically in geometric topology, the Kirby–Siebenmann class is an obstruction for topological manifolds to allow a PL-structure.

In algebraic topology, cobordism theories are fundamental extraordinary cohomology theories, and categories of cobordisms are the domains of topological quantum field theories.

While the precise function of CD133 remains unknown, it has been proposed that it acts as an organizer of cell membrane topology.

In differential geometry the Hitchin-Thorpe inequality is a relation which restricts the topology of 4-manifolds that carry an Einstein metric.

Mikhail Mikhailovich Postnikov (; 27 October 1927 – 27 May 2004) was a Soviet mathematician, known for his work in algebraic and differential topology.

Jean Leray (; 7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology.

Convergence in this topology is a crucial ingredient of hyperbolic Dehn surgery, a fundamental tool in the theory of hyperbolic 3-manifolds.

In mathematics, the Chabauty topology is a certain topological structure introduced in 1950 by Claude Chabauty, on the set of all closed subgroups of a locally compact group G. The intuitive idea may be seen in the case of the set of all lattices in a Euclidean space E. There these are only certain of the closed subgroups: others can be found by in a sense taking limiting cases or degenerating a certain sequence of lattices. One can find linear subspaces or discrete groups that are lattices in a subspace, depending on how one takes a limit. This phenomenon suggests that the set of all closed subgroups carries a useful topology. This topology can be derived from the Vietoris topology construction, a topological structure on all non-empty subsets of a space.

When talking about spaces with more structure than just topology, like topological groups, the natural meaning of the words “completely metrizable” would arguably be the existence of a complete metric that is also compatible with that extra structure, in addition to inducing its topology. For abelian topological groups and topological vector spaces, “compatible with the extra structure” might mean that the metric is invariant under translations. However, no confusion can arise when talking about an abelian topological group or a topological vector space being completely metrizable: it can be proven that every abelian topological group (and thus also every topological vector space) that is completely metrizable as a topological space (i. e., admits a complete metric that induces its topology) also admits an invariant complete metric that induces its topology.

Moreover, its mathematical foundation is also of theoretical importance. The unique features of TDA make it a promising bridge between topology and geometry.

In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms.

With Michael Aschbacher and Bob Oliver, she is an author of the book Fusion Systems in Algebra and Topology (Cambridge University Press, 2011).

Insteon technology uses a dual-mesh networking topology in which all devices are peers and each device independently transmits, receives, and repeats messages.

Thomas Schick, Oberwolfach 2012 Thomas Schick (born 22 May 1969 in Alzey) is a German mathematician, specializing in algebraic topology and differential geometry.

Garnica et al. and Binder et al. recovered a similar topology with Pluteus, Volvariella and Melanoleuca as a monophyletic group. Justo et al.

3, 1994, pp. 41–50 ; B. Wiest and M. T. Green: A natural framing of knots. In: Geometry & Topology. v. 2, 1998, pp.

The main aim of topology control in this domain is to save energy, reduce interference between nodes and extend lifetime of the network.

Hopf link Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry.

Individual nodes use the topology information to compute next hop paths regard to all nodes in the network using shortest hop forwarding paths.

Peter B. Shalen (born c. 1946) is an American mathematician, working primarily in low-dimensional topology. He is the "S" in JSJ decomposition.

Věra Šedivá-Trnková (March 16, 1934 – 27 May 2018) was a Czech mathematician known for her work in topology and in category theory.

Georgii Dmitrievic Suvorov (19 May 1919 – 12 October 1984) was a Russian mathematician who made major contributions to the function theory and topology.

The study of network topology recognizes eight basic topologies: point-to-point, bus, star, ring or circular, mesh, tree, hybrid, or daisy chain.

Observe that if its shown that is a Banach space then will be a Banach disk in any TVS that contains as a bounded subset. This is because the Minkowski functional is defined in purely algebraic terms. Consequently, the question of whether or not forms a Banach space is dependent only on the disk and the Minkowski functional , and not on any particular TVS topology that may carry. Thus the requirement that a Banach disk in a TVS be a bounded subset of is the only property that ties a Banach disk's topology to the topology of its containing TVS .

Example topology of a Fibre Channel switched fabric network A storage area network built with two separate switched fabrics (red and blue) to increase reliability. In the Fibre Channel Switched Fabric (FC-SW-6) topology, devices are connected to each other through one or more Fibre Channel switches. While this topology has the best scalability of the three FC topologies (the other two are Arbitrated Loop and point-to-point), it is the only one requiring switches, which are costly hardware devices. Visibility among devices (called nodes) in a fabric is typically controlled with Fibre Channel zoning.

Sierpiński space, then Scott-continuous functions are characteristic functions, and thus Sierpiński space is the classifying topos for open sets. A subset O of a partially ordered set P is called Scott-open if it is an upper set and if it is inaccessible by directed joins, i.e. if all directed sets D with supremum in O have non-empty intersection with O. The Scott-open subsets of a partially ordered set P form a topology on P, the Scott topology. A function between partially ordered sets is Scott-continuous if and only if it is continuous with respect to the Scott topology.

The open sets in a given topological space when ordered by inclusion form a lattice on which the Scott topology can be defined. A subset X of a topological space T is compact with respect to the topology on T (in the sense that every open cover of X contains a finite subcover of X) if and only if the set of open neighbourhoods of X is open with respect to the Scott topology. For CPO, the cartesian closed category of dcpo's, two particularly notable examples of Scott-continuous functions are curry and apply. (See theorems 1.2.

Holotypes of O. simus (A and C) and Tropeognathus mesembrinus (B and D) A topology made by Andres and Myers in 2013 placed Ornithocheirus within the family Ornithocheiridae in a more derived position than Tropeognathus, but in a more basal position than Coloborhynchus, and the family itself is placed within the more inclusive clade Ornithocheirae. In 2019, the description of the new ornithocheirid Ferrodraco by Pentland et al. had made paleontologists reclassify the family Ornithocheiridae. In the analysis, a topology placed Ornithocheirus within the subfamily Ornithocheirinae in a more derived position than Coloborhynchus, which contradicts the anterior topology by Andres & Myers in 2013.

It is assumed that the system is initially in a legitimate configuration. Then the network topology is changed; the superstabilization time is the maximum time it takes for the system to reach a legitimate configuration again. Similarly, the adjustment measure is the maximum number of nodes that have to change their state after such changes. The “almost-legitimate configurations” which occur after one topology change can be formally modelled by using passage predicates: a passage predicate is a predicate that holds after a single change in the network topology, and also during the convergence to a legitimate configuration.

Convergence is the state of a set of routers that have the same topological information about the internetwork in which they operate. For a set of routers to have converged, they must have collected all available topology information from each other via the implemented routing protocol, the information they gathered must not contradict any other router's topology information in the set, and it must reflect the real state of the network. In other words: In a converged network all routers "agree" on what the network topology looks like. Convergence is an important notion for a set of routers that engage in dynamic routing.

The topology of a β-sheet describes the order of hydrogen-bonded β-strands along the backbone. For example, the flavodoxin fold has a five-stranded, parallel β-sheet with topology 21345; thus, the edge strands are β-strand 2 and β-strand 5 along the backbone. Spelled out explicitly, β-strand 2 is H-bonded to β-strand 1, which is H-bonded to β-strand 3, which is H-bonded to β-strand 4, which is H-bonded to β-strand 5, the other edge strand. In the same system, the Greek key motif described above has a 4123 topology.

The intuitive meaning of a stack is that it is a fibred category such that "all possible gluings work". The specification of gluings requires a definition of coverings with regard to which the gluings can be considered. It turns out that the general language for describing these coverings is that of a Grothendieck topology. Thus a stack is formally given as a fibred category over another base category, where the base has a Grothendieck topology and where the fibred category satisfies a few axioms that ensure existence and uniqueness of certain gluings with respect to the Grothendieck topology.

Any two distinct points in [-1,1] are topologically distinguishable under the overlapping interval topology as one can always find an open set containing one but not the other point. However, every non-empty open set contains the point 0 which can therefore not be separated from any other point in [-1,1], making [-1,1] with the overlapping interval topology an example of a T0 space that is not a T1 space. The overlapping interval topology is second countable, with a countable basis being given by the intervals [-1,s), (r,s) and (r,1] with r < 0 < s and r and s rational.

Bendsøe and Kikuchi can be consider as a pioneers in the microstructure topology optimization domain and was originally intended for the design of mechanical structures . In additive manufacturing, topology optimization is considered as an efficient method to find the right topologies by running optimization algorithms targeting to achieve the prescribed properties . The effective properties of the material structures are found unsing a numeriacal homogenization method, and then the topology optimization is applied to find the best distribution of material phases that extremizes the objective function . Inverse homogenization method has been introduced to achieve specific macro-scale behaviors with desired Poisson’s ratio.

SPB can use many VIDs, agreeing on which VIDs are used for which purposes. The IIH PDUs carry a digest of all the used VIDs, referred to as the Multiple Spanning Tree Configuration TLV which uses a common and compact encoding reused from 802.1Q. For the purposes of loop prevention SPB neighbors may also support a mechanism to verify that the contents of their topology databases are synchronized. Exchanging digests of SPB topology information, using the optional SPB-Digest sub-TLV, allows nodes to compare information and take specific action where a mismatch in topology is indicated.

In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis. One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.) if they are closed (respectively, compact, etc.) with respect to the weak topology.

To detect sets of genes that fit poorly to the reference tree, one can use statistical tests of topology, such as the Kishino–Hasegawa (KH), Shimodaira–Hasegawa (SH), and Approximately Unbiased (AU) tests. These tests assess the likelihood of the gene sequence alignment when the reference topology is given as the null hypothesis. The rejection of the reference topology is an indication that the evolutionary history for that gene family is inconsistent with the reference tree. When these inconsistencies cannot be explained using a small number of non-horizontal events, such as gene loss and duplication, an HGT event is inferred.

FDDI provides a 100 Mbit/s optical standard for data transmission in local area network that can extend in range up to . Although FDDI logical topology is a ring-based token network, it did not use the IEEE 802.5 token ring protocol as its basis; instead, its protocol was derived from the IEEE 802.4 token bus timed token protocol. In addition to covering large geographical areas, FDDI local area networks can support thousands of users. FDDI offers both a Dual-Attached Station (DAS), counter- rotating token ring topology and a Single-Attached Station (SAS), token bus passing ring topology.

In biology literature, the term topology is also used to refer to mutual orientation of regular secondary structures, such as alpha-helices and beta strands in protein structure . For example, two adjacent interacting alpha-helices or beta-strands can go in the same or in opposite directions. Topology diagrams of different proteins with known three-dimensional structure are provided by PDBsum (an example).

For , the image in of under form a neighborhood basis at . This is, by the way it is constructed, a neighborhood basis both in the group topology and the relative topology. Since multiplication in is analytic, the left and right translates of this neighborhood basis by a group element gives a neighborhood basis at . These bases restricted to gives neighborhood bases at all .

Much like Winged Edge, quad-edge structures are used in programs to store the topology of a 2D or 3D polygonal mesh. The mesh itself does not need to be closed in order to form a valid quad-edge structure. Using a quad-edge structure, iterating through the topology is quite easy. Often, the interface to quad- edge topologies is through directed edges.

There are several notable interesting features of the recent applications of TDA: # Combining tools from several branches of mathematics. Besides the obvious need for algebra and topology, partial differential equations, algebraic geometry, representation theory, statistics, combinatorics, and Riemannian geometry have all found use in TDA. # Quantitative analysis. Topology is considered to be very soft since many concepts are invariant under homotopy.

Editorial Board, Annals of Mathematics. Accessed February 8, 2010 Currently he is an Editorial Board member for Duke Mathematical Journal,Duke Mathematical JournalGeometric and Functional Analysis,Editorial Board, Geometric and Functional Analysis. Accessed February 8, 2010 the Journal of Topology and Analysis,Editorial Board. Journal of Topology and Analysis. Accessed February 8, 2010 Groups, Geometry and Dynamics,Editorial Board, Groups, Geometry and Dynamics.

Finite-dimensional vector spaces over local fields and division algebras under the topology uniquely determined by the field's topology are studied, and lattices are defined topologically, an analogue of Minkowski's theorem is proved in this context, and the main theorems about character groups of these vector spaces, which in the commutative one-dimensional case reduces to `self duality’ for local fields, are shown.

TopoARTMarko Tscherepanow. (2010) TopoART: A Topology Learning Hierarchical ART Network, In: Proceedings of the International Conference on Artificial Neural Networks (ICANN), Part III, LNCS 6354, 157-167 combines fuzzy ART with topology learning networks such as the growing neural gas. Furthermore, it adds a noise reduction mechanism. There are several derived neural networks which extend TopoART to further learning paradigms.

In 1942–45, Samuel Eilenberg and Saunders Mac Lane introduced categories, functors, and natural transformations as part of their work in topology, especially algebraic topology. Their work was an important part of the transition from intuitive and geometric homology to homological algebra. Eilenberg and Mac Lane later wrote that their goal was to understand natural transformations. That required defining functors, which required categories.

In mathematics, a topological space X is called countably generated if the topology of X is determined by the countable sets in a similar way as the topology of a sequential space (or a Fréchet space) by the convergent sequences. The countable generated spaces are precisely the spaces having countable tightness - therefore the name countably tight is used as well.

His research is on algebraic topology and differential topology. In work with Ib Madsen, he resolved the Mumford Conjecture about rational characteristic classes of surface bundles in the limit as the genus tends to infinity.Allen Hatcher, A Short Exposition of the Madsen–Weiss Theorem Building on earlier work of Thomas Goodwillie, he developed Embedding Calculus, a Calculus of functors for embeddings of manifolds.

Robert Ernest Gompf (born 1957) is an American mathematician specializing in geometric topology. Gompf received a Ph.D. in 1984 from the University of California, Berkeley under the supervision of Robion Kirby (An invariant for Casson handles, disks and knot concordants). He is now a professor at the University of Texas at Austin. His research concerns the topology of 4-manifolds.

The physical network topology can be directly represented in a network diagram, as it is simply the physical graph represented by the diagrams, with network nodes as vertices and connections as undirected or direct edges (depending on the type of connection). The logical network topology can be inferred from the network diagram if details of the network protocols in use are also given.

It has been shown that Lefschetz pencils exist in characteristic zero. They apply in ways similar to, but more complicated than, Morse functions on smooth manifolds. It has also been shown that Lefschetz pencils exist in characteristic p for the étale topology. Simon Donaldson has found a role for Lefschetz pencils in symplectic topology, leading to more recent research interest in them.

Kathryn Hess (born 1967) is a professor of mathematics at École Polytechnique Fédérale de Lausanne (EPFL) and is known for her work on homotopy theory, category theory, and algebraic topology, both pure and applied. In particular, she applies the methods of algebraic topology to better understanding neurology, cancer biology, and materials science. She is a fellow of the American Mathematical Society.

Semantic P2P networks are a new type of P2P network. It combines the advantages of unstructured P2P networks and structural P2P networks, and avoids their disadvantages. In Semantic P2P networks, nodes are classified as DNS-like domain names with semantic meanings such as Alice @Brittney.popular.music. Semantic P2P networks contains prerequisite virtual tree topology and net-like topology formed by cached nodes.

The structure, or "topology" of a grid can vary depending on the constraints of budget, requirements for system reliability, and the load and generation characteristics. The physical layout is often forced by what land is available and its geology. Distribution networks are divided into two types, radial or network. The simplest topology for a distribution or transmission grid is a radial structure.

On 5 November 1923 he was elected a Fellow of St John's. He worked on the foundations of combinatorial topology, and proposed that a notion of equivalence be defined using only three elementary "moves". Newman's definition avoided difficulties that had arisen from previous definitions of the concept. Publishing over twenty papers established his reputation as an "expert in modern topology".

Given the closed interval [-1,1] of the real number line, the open sets of the topology are generated from the half-open intervals [-1,b) and (a,1] with a < 0 < b. The topology therefore consists of intervals of the form [-1,b), (a,b), and (a,1] with a < 0 < b, together with [-1,1] itself and the empty set.

Accessed September 14, 2017 She is currently an Associate Editor of the Journal of the American Mathematical Society,Editorial Board, Journal of the American Mathematical Society. Accessed September 14, 2017. an Editorial Board member Geometry & Topology Monographs book series,Editorial Board, Geometry & Topology Monographs. Accessed September 14, 2017 and a Consulting Editor for the Proceedings of the Edinburgh Mathematical Society.

Brian Hayward Bowditch (born 1961Brian H. Bowditch: Me. Bowditch's personal information page at the University of Warwick) is a British mathematician known for his contributions to geometry and topology, particularly in the areas of geometric group theory and low-dimensional topology. He is also known for solving the angel problem. Bowditch holds a chaired Professor appointment in Mathematics at the University of Warwick.

On Rn or Cn, the closed sets of the Zariski topology are the solution sets of systems of polynomial equations. A linear graph has a natural topology that generalises many of the geometric aspects of graphs with vertices and edges. The Sierpiński space is the simplest non-discrete topological space. It has important relations to the theory of computation and semantics.

For any algebraic objects we can introduce the discrete topology, under which the algebraic operations are continuous functions. For any such structure that is not finite, we often have a natural topology compatible with the algebraic operations, in the sense that the algebraic operations are still continuous. This leads to concepts such as topological groups, topological vector spaces, topological rings and local fields.

A closed ball of radius r is a closed r-ball. Every closed ball is a closed set in the topology induced on M by d. Note that the closed ball D(x; r) might not be equal to the closure of the open ball B(x; r). ;Closed set: A set is closed if its complement is a member of the topology.

In 3-dimensional topology, a part of the mathematical field of geometric topology, the Casson invariant is an integer-valued invariant of oriented integral homology 3-spheres, introduced by Andrew Casson. Kevin Walker (1992) found an extension to rational homology 3-spheres, called the Casson–Walker invariant, and Christine Lescop (1995) extended the invariant to all closed oriented 3-manifolds.

A topological vector space is a topological module over a topological field. An abelian topological group can be considered as a topological module over Z, where Z is the ring of integers with the discrete topology. A topological ring is a topological module over each of its subrings. A more complicated example is the I-adic topology on a ring and its modules.

Occasionally, one needs to use the tools of topology but a "set of points" is not available. In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are structures defined on arbitrary categories that allow the definition of sheaves on those categories, and with that the definition of general cohomology theories.

The reason is quite clear. The accuracy of a centrality measure depends on network topology, but complex networks have heterogenous topology. Hence a centrality measure which is appropriate for identifying highly influential nodes will most likely be inappropriate for the remainder of the network. This has inspired the development of novel methods designed to measure the influence of all network nodes.

The initial topology on X can be characterized by the following characteristic property: A function g from some space Z to X is continuous if and only if f_i \circ g is continuous for each i ∈ I. Characteristic property of the initial topology Note that, despite looking quite similar, this is not a universal property. A categorical description is given below.

In a distributed bus network, all of the nodes of the network are connected to a common transmission medium with more than two endpoints, created by adding branches to the main section of the transmission medium – the physical distributed bus topology functions in exactly the same fashion as the physical linear bus topology because all nodes share a common transmission medium.

His interests were in set theory and general topology. He found necessary and sufficient conditions for metrizability and orderability of pseudometric and ultrametric spaces.

William Robert "Red" Alford (July 21, 1937 – May 29, 2003) was an American mathematician who worked in the fields of topology and number theory.

The fabric topology allows the connection of up to the theoretical maximum of 16 million devices, limited only by the available address space (224).

Kiyoshi Igusa (born November 28, 1949) is a Japanese-American mathematician and a professor at Brandeis University. He works in representation theory and topology.

There are two natural types of functors between sites. They are given by functors that are compatible with the topology in a certain sense.

String topology, a branch of mathematics, is the study of algebraic structures on the homology of free loop spaces. The field was started by .

16 January 2011. “Universe as Doughnut: New Data, New Debate” The theory describes the shape of the universe (topology) as a three-dimensional torus.

Wacław Bolesław Marzantowicz is a Polish mathematician known for his contributions in the number theory and topology, President of Polish Mathematical Society (2014–2019).

Jason Alan Behrstock is a mathematician at City University of New York known for his work in geometric group theory and low-dimensional topology.

In algebraic topology, the mapping spectrum F(X, Y) of spectra X, Y is characterized by :[X \wedge Y, Z] = [X, F(Y, Z)].

The extreme points of a compact convex form a Baire space (with the subspace topology) but this set may fail to be closed in .

Oberwolfach in 2006 Wolfgang Lück (born 19 February 1957 in Herford) is a German mathematician who is an internationally recognized expert in Algebraic topology.

In algebraic topology, Johnson–Wilson theory E(n) is a generalized cohomology theory introduced by . Real Johnson–Wilson theory ER(n) was introduced by .

As is true of other members of the PTS-GFL superfamily, the IIC domains of these permeases probably have a uniform 10 TMS topology.

In mathematics, specifically geometry and topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain.

Herbert Seifert Herbert Karl Johannes Seifert (; 27 May 1907, Bernstadt – 1 October 1996, Heidelberg) was a German mathematician known for his work in topology.

AS3356 consistently had one of the highest ranked connectivity degrees on the Internet.Visualizing Internet Topology at a Macroscopic Scale January 2009, caida.orgAS ranking caida.

Surprisingly, TopPred predicts a 12 TMS topology for the yeast Pho89 protein, but the homologous regions are not predicted to show similar topological features.

A subset of X is closed/open if and only if its preimage under fi is closed/open in Y_i for each i ∈ I. The final topology on X can be characterized by the following characteristic property: a function g from X to some space Z is continuous if and only if g \circ f_i is continuous for each i ∈ I. Characteristic property of the final topology By the universal property of the disjoint union topology we know that given any family of continuous maps fi : Yi -> X, there is a unique continuous map :f\colon \coprod_i Y_i \to X. If the family of maps fi covers X (i.e. each x in X lies in the image of some fi) then the map f will be a quotient map if and only if X has the final topology determined by the maps fi.

Oberwolfach, 1987 Horst Herrlich (11 September 1937, in Berlin – 13 March 2015, in Bremen) was a German mathematician, known as a pioneer of categorical topology.

For the best Caching performance, a dedicated topology is recommended because the role instances do not share their resources with other application code and services.

Its product capabilities included network discovery, topology mapping, and root cause analysis (RCA) utilizing graph theory.Ellen Messmer, Network World. "Nimsoft acquires the assets of Cittio".

Lev Genrikhovich Schnirelmann (also Shnirelman, Shnirel'man; ; January 2, 1905 – September 24, 1938) was a Soviet mathematician who worked on number theory, topology and differential geometry.

Bulletin of the London Mathematical Society vol. 37 (2005), pp. 459–466Francois Dahmani, Combination of convergence groups. Geometry and Topology, Volume 7 (2003), 933–963I.

Frank Stringfellow Quinn, III (born 1946) is an American mathematician and professor of mathematics at Virginia Polytechnic Institute and State University, specializing in geometric topology.

Franklin Paul Peterson (1930–2000) was an American mathematician specializing in algebraic topology. He was a professor of mathematics at the Massachusetts Institute of Technology...

Bourbaki and Algebraic Topology by John McCleary (PDF) gives documentation (translated into English from French originals). Algebraic homology remains the primary method of classifying manifolds.

This example shows that a translation-invariant (pseudo)metric is not enough to guarantee a vector topology, which leads us to define paranorms and -seminorms.

Christos Dimitriou Papakyriakopoulos (), commonly known as Papa (Greek: Χρήστος Δημητρίου Παπακυριακόπουλος ; June 29, 1914 – June 29, 1976), was a Greek mathematician specializing in geometric topology.

Respect to the topology of the given space of generalized functions. They are also known as Friedrichs mollifiers after Kurt Otto Friedrichs, who introduced them.See .

This is a list of particular manifolds, by Wikipedia page. See also list of geometric topology topics. For categorical listings see :Category:Manifolds and its subcategories.

Mónica Clapp is a mathematician at the Universidad Nacional Autónoma de México (UNAM) known for her work in nonlinear partial differential equations and algebraic topology.

Möbius ladders have also been used as the shape of a superconducting ring in experiments to study the effects of conductor topology on electron interactions.; .

In the mathematical field of topology, a manifold M is called topologically rigid if every manifold homotopically equivalent to M is also homeomorphic to M.

Sir Erik Christopher Zeeman FRS (4 February 1925 – 13 February 2016), was a British mathematician, known for his work in geometric topology and singularity theory.

Edgar Henry Brown, Jr. (born 27 December 1926) is an American mathematician specializing in algebraic topology, and for many years a professor at Brandeis University.

On a theorem of Kontsevich. Algebraic and Geometric Topology, vol. 3 (2003), pp. 1167–1224James Conant, and Karen Vogtmann, Infinitesimal operations on complexes of graphs.

Ronald Alan Fintushel (born 1945) is an American mathematician, specializing in low-dimensional geometric topology (specifically of 4-manifolds) and the mathematics of gauge theory.

In geometric topology, the side-approximation theorem was proved by . It implies that a 2-sphere in R3 can be approximated by polyhedral 2-spheres.

Link Layer Topology Discovery (LLTD) is a proprietary link layer protocol for network topology discovery and quality of service diagnostics. Microsoft developed it as part of the Windows Rally set of technologies. The LLTD protocol operates over both wired (such as Ethernet (IEEE 802.3) or power line communication) as well as wireless networks (such as IEEE 802.11). LLTD is included in Windows Vista and Windows 7.

The first algorithm over all fields for persistent homology in algebraic topology setting was described by Barannikov through reduction to the canonical form by upper-triangular matrices. The first algorithm for persistent homology over F_2 was given by Edelsbrunner et al. Zomorodian and Carlsson gave the first practical algorithm to compute persistent homology over all fields. Edelsbrunner and Harer's book gives general guidance on computational topology.

The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions. By virtue of their order, the rationals carry an order topology. The rational numbers, as a subspace of the real numbers, also carry a subspace topology.

A Scott-continuous function is always monotonic. A subset of a partially ordered set is closed with respect to the Scott topology induced by the partial order if and only if it is a lower set and closed under suprema of directed subsets. A directed complete partial order (dcpo) with the Scott topology is always a Kolmogorov space (i.e., it satisfies the T0 separation axiom).

Given a ring R and an R-module M, a descending filtration of M is a decreasing sequence of submodules M_n. This is therefore a special case of the notion for groups, with the additional condition that the subgroups be submodules. The associated topology is defined as for groups. An important special case is known as the I-adic topology (or J-adic, etc.).

There is a natural projection : \pi : TM \twoheadrightarrow M defined by \pi(x, v) = x. This projection maps each tangent space T_xM to the single point x . The tangent bundle comes equipped with a natural topology (described in a section below). With this topology, the tangent bundle to a manifold is the prototypical example of a vector bundle (a fiber bundle whose fibers are vector spaces).

Helen Frances Cullen (January 4, 1919 – August 25, 2007) was an American mathematician specializing in topology. She worked for many years as a professor of mathematics at the University of Massachusetts Amherst and was the first female faculty member in the mathematics department at Amherst. She was known as the author of the book Introduction to General Topology (Heath, 1968), as well as for her outspoken antisemitism.

In mathematics, homotopy theory is a systematic study of situations in which maps come with homotopies between them. It originated as a topic in algebraic topology but nowadays it is studied as an independent discipline. Besides algebraic topology, the theory has also been in used in other areas of mathematics such as algebraic geometry (e.g., A¹ homotopy theory) and category theory (specifically the study of higher categories).

Oberwolfach (2000) Matthias Kreck (born 22 July 1947 in Dillenburg) is a German mathematician who works in the areas of Algebraic Topology and Differential topology. From 1994 to 2002 he was director of the Mathematical Research Institute of Oberwolfach and from October 2006 to September 2011 he was the director of the Hausdorff Center for Mathematics at the University of Bonn, where he is currently a professor.

Informally distributed, 1969. Notes, December 1969, Oxford Univ. Introduced November 1969, Dana Scott's untyped set theoretic model constructed a proper topology for any λ-calculus model whose function space is limited to continuous functions. The result of a Scott continuous λ-calculus topology is a function space built upon a programming semantic allowing fixed point combinatorics, such as the Y combinator, and data types.

In algebraic topology, simplicial complexes are often useful for concrete calculations. For the definition of homology groups of a simplicial complex, one can read the corresponding chain complex directly, provided that consistent orientations are made of all simplices. The requirements of homotopy theory lead to the use of more general spaces, the CW complexes. Infinite complexes are a technical tool basic in algebraic topology.

In design mode components are linked to construct a topology. Linked components enable a signal flow creating a pipe filter machine. When a signal is set on a component, it filters the signal in some way and the filtered signal can then be piped to the next component in the linked chain of components that form the topology. The components can be either static or adaptive.

Fig 2: Topology of the very-sparse matrix. Characteristics of the Very Sparse Matrix Converter topology are 12 Transistors, 30 Diodes, and 10 Isolated Driver Potentials. There are no limitations in functionality compared to the Direct Matrix Converter and Sparse Matrix Converter. Compared to the Sparse Matrix Converter there are fewer transistors but higher conduction losses due to the increased number of diodes in the conduction paths.

Since each vulcano node has complete network topology available, routing is just a matter of matching flare keys with nodes key-space and find the node holding the flare. Network topology is also saved in the distributed directory inside magma filesystem. Vulcano nodes can periodically check their information against contents of directory to know if something has changed. Nodes also periodically save their own information inside directory.

ONE-NET supports star, peer-to-peer and multi-hop topology. Star network topology can be used to lower complexity and cost of peripherals, and also simplifies encryption key management. In peer-to-peer mode, a master device configures and authorizes peer-to-peer transactions. Employing repeaters and a configurable repetition radius multi-hop mode allows to cover larger areas or route around dead areas.

Strictly speaking, replacing a component with one of an entirely different type is still the same topology. In some contexts, however, these can loosely be described as different topologies. For instance, interchanging inductors and capacitors in a low-pass filter results in a high-pass filter. These might be described as high-pass and low-pass topologies even though the network topology is identical.

His Ph.D. students include Martin Arkowitz, Robert Lewis, Jean-Pierre Meyer, and Norman Stein. In 1962 Olum initiated the Cornell Topology Festival, an annual regional mathematics conference.Cornell Topology Festival From 1963 to 1966, Olum served as Mathematics Department chair, and recruited a number of talented faculty. Olum advocated the abolition of the House Committee on Unamerican Activities,Cornell Daily Sun, March 3, 1961 p.

The conserved GMN motif is at the outside end of the first TM domain, and when the glycine (G) in this motif was mutated to a cysteine (C) in Mrs2p, Mg2+ transport was strongly reduced. The TM topology of the MRS2 and LPE10 proteinsThe figure shows the experimentally determined topology of Mrs2p and Lpe10p as adapted from Bui et al. (1999) and Gregan et al. (2001a).

However, the initiation sequences are generally either AT-rich or exhibit bent or curved DNA topology. The ORC4 protein is known to bind the AT-rich portion of the origin of replication in S. pombe using AT hook motifs. The mechanism of origin recognition in higher eukaryotes is not well understood but it is thought that the ORC1-6 proteins depend on unusual DNA topology for binding.

Geometry & Topology is a peer-refereed, international mathematics research journal devoted to geometry and topology, and their applications. It is currently based at the University of Warwick, United Kingdom, and published by Mathematical Sciences Publishers, a nonprofit academic publishing organisation. It was founded in 1997Allyn Jackson, The slow revolution of the free electronic journal, Notices of the American Mathematical Society, vol. 47 (2000), no.

ControlNet cables consist of RG-6 coaxial cable with BNC connectors, though optical fiber is sometimes used for long distances. The network topology is a bus structure with short taps. ControlNet also supports a star topology if used with the appropriate hardware. ControlNet can operate with a single RG-6 coaxial cable bus, or a dual RG-6 coaxial cable bus for cable redundancy.

FireWire can connect up to 63 peripherals in a tree or daisy-chain topology (as opposed to Parallel SCSI's electrical bus topology). It allows peer-to-peer device communication — such as communication between a scanner and a printer — to take place without using system memory or the CPU. FireWire also supports multiple hosts per bus. It is designed to support plug and play and hot swapping.

Below is cladogram following a topology by Andres, Clark, and Xu (2014). They included three subfamilies within the Ctenochasmatidae: Ctenochasmatinae, Gnathosaurinae and Moganopterinae, while also including several basal genera. In 2018, Longrich, Martill, and Andres recovered a very similar topology to the one by Andres, Clark, and Xu (2014), but they recovered more genera within the family, as shown below.Longrich, N.R., Martill, D.M., and Andres, B. (2018).

An important special case is where the topology on X is the discrete topology; in this case X can be identified with a cardinal number κ and C(X) with the real algebra Rκ of functions from κ to R. The hyperreal fields we obtain in this case are called ultrapowers of R and are identical to the ultrapowers constructed via free ultrafilters in model theory.

One of the first non-examples comes from the induced topology of the rationals inside of the real line with the standard euclidean topology. Given an indexing of the rationals by the natural numbers , so a bijection , and let where }, which is an open, dense subset in . Then, because the intersection of every open set in is empty, the space cannot be a Baire space.

Topology Change Notification (TCN) BPDUs are used to inform other switches of port changes. TCNs are injected into the network by a non-root switch and propagated to the root. Upon receipt of the TCN, the root switch will set the Topology Change flag in its normal BPDUs. This flag is propagated to all other switches and instructs them to rapidly age out their forwarding table entries.

Gunnar E. Carlsson (born August 22, 1952 in Stockholm, Sweden) is an American mathematician, working in algebraic topology. He is known for his work on the Segal conjecture, and for his work on applied algebraic topology, especially topological data analysis. He is a Professor Emeritus in the Department of Mathematics at Stanford University. He is the founder and president of the predictive technology company Ayasdi.

Gas Network Topology In the gas networks simulation and analysis, matrices turned out to be the natural way of expressing the problem. Any network can be described by set of matrices based on the network topology. Consider the gas network by the graph below. The network consists of one source node (reference node) L1, four load nodes (2, 3, 4 and 5) and seven pipes or branches.

Suppose that (X,Y,b) is a pairing of vector spaces over . :Notation: For all , let denote the linear functional on defined by and let . Similarly, for all , let be defined by and let . The weak topology on induced by Y (and ) is the weakest TVS topology on , denoted by \sigma(X,Y,b) or simply \sigma(X,Y), making all maps continuous, as y ranges over .

Rokhlin's contributions to topology include Rokhlin's theorem, a result of 1952 on the signature of 4-manifolds. He also worked in the theory of characteristic classes, homotopy theory, cobordism theory, and in the topology of real algebraic varieties. In measure theory, Rokhlin introduced what are now called Rokhlin partitions. He introduced the notion of standard probability space, and characterised such spaces up to isomorphism mod 0\.

Sigma algebras are a special case of a topology, and so thereby allow notions such as continuous and differentiable functions to be defined. These are the basic ingredients to a dynamical system: a phase space, a topology (sigma algebra) on that space, a measure, and an invertible function providing the time evolution. Conservative systems are those systems that do not shrink their phase space over time.

The transition from "Euclidean" to "topological" is forgetful. Topology distinguishes continuous from discontinuous, but does not distinguish rectilinear from curvilinear. Intuition tells us that the Euclidean structure cannot be restored from the topology. A proof uses an automorphism of the topological space (that is, self-homeomorphism) that is not an automorphism of the Euclidean space (that is, not a composition of shifts, rotations and reflections).

Both transitions are not surjective, that is, not every B-space results from some A-space. First, a 3-dim Euclidean space is a special (not general) case of a Euclidean space. Second, a topology of a Euclidean space is a special case of topology (for instance, it must be non-compact, and connected, etc). We denote surjective transitions by a two-headed arrow, "↠" rather than "→".

Libgober's early work studies the diffeomorphism type of complete intersections in complex projective space. This later led to the discovery of relations between Hodge and Chern numbers.A.Libgober, J.Wood, Differentiable structures on complete intersections I, Topology, 21 (1982),469-482 He introduced the technique of Alexander polynomialA.Libgober,Development of the theory of Alexander invariants in algebraic geometry, Topology of algebraic varieties and singularities, 3–17, Contemp. Math.

The inter-node communication is based on FDR InfiniBand. The topology of the InfiniBand network is a two-dimensional hyper-crossbar. This means that a two-dimensional mesh of InfiniBand switches is built, and the two InfiniBand ports of a node are connected to one switch in each of the dimensions. The hyper-crossbar topology was first introduced by the Japanese CP-PACS collaboration of particle physicists.

Since the bus topology consists of only one wire it is less expensive to implement than other topologies, but the savings are offset by the higher cost of managing the network. Additionally, since the network is dependent on the single cable, it can be the single point of failure of the network. In this topology data being transferred may be accessed by any node.

However, a tree network connected to another tree network is still topologically a tree network, not a distinct network type. A hybrid topology is always produced when two different basic network topologies are connected. A star-ring network consists of two or more ring networks connected using a multistation access unit (MAU) as a centralized hub. Snowflake topology is a star network of star networks.

From the point of view of topology, the seminormed vector space that we started with has a lot of extra structure; for example, it is a vector space, and it has a seminorm, and these define a pseudometric and a uniform structure that are compatible with the topology. Also, there are several properties of these structures; for example, the seminorm satisfies the parallelogram identity and the uniform structure is complete. The space is not T0 since any two functions in L2(R) that are equal almost everywhere are indistinguishable with this topology. When we form the Kolmogorov quotient, the actual L2(R), these structures and properties are preserved.

One often considers continuous group actions: the group G is a topological group, X is a topological space, and the map is continuous with respect to the product topology of . The space X is also called a G-space in this case. This is indeed a generalization, since every group can be considered a topological group by using the discrete topology. All the concepts introduced above still work in this context, however we define morphisms between G-spaces to be continuous maps compatible with the action of G. The quotient X/G inherits the quotient topology from X, and is called the quotient space of the action.

The real numbers form a topological group under addition In mathematics, a topological group is a group together with a topology on such that both the group's binary operation and the function mapping group elements to their respective inverses are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology. Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics.

A more correct term for these classes of object (that is, a network where the type of component is specified but not the absolute value) is prototype network. Electronic network topology is related to mathematical topology, in particular, for networks which contain only two-terminal devices, circuit topology can be viewed as an application of graph theory. In a network analysis of such a circuit from a topological point of view, the network nodes are the vertices of graph theory and the network branches are the edges of graph theory. Standard graph theory can be extended to deal with active components and multi-terminal devices such as integrated circuits.

In August 2008, a team of computer scientists at UCSD published a scalable design for network architecture that uses a topology inspired by the fat tree topology to realize networks that scale better than those of previous hierarchical networks. The architecture uses commodity switches that are cheaper and more power-efficient than high-end modular data center switches. This topology is actually a special instance of a Clos network, rather than a fat-tree as described above. That is because the edges near the root are emulated by many links to separate parents instead of a single high-capacity link to a single parent.

A topological space X is said to be locally normal if and only if each point, x, of X has a neighbourhood that is normal under the subspace topology. Note that not every neighbourhood of x has to be normal, but at least one neighbourhood of x has to be normal (under the subspace topology). Note however, that if a space were called locally normal if and only if each point of the space belonged to a subset of the space that was normal under the subspace topology, then every topological space would be locally normal. This is because, the singleton {x} is vacuously normal and contains x.

While the 802.1Qav FQTSS/CBS works very well with soft real-time traffic, worst-case delays are both hop count and network topology dependent. Pathological topologies introduce delays, so buffer size requirements have to consider network topology. IEEE 802.1Qch Cyclic Queuing and Forwarding (CQF), also known as the Peristaltic Shaper (PS), introduces double buffering which allows bridges to synchronize transmission (frame enqueue/dequeue operations) in a cyclic manner, with bounded latency depending only on the number of hops and the cycle time, completely independent of the network topology. CQF can be used with the IEEE 802.1Qbv time-aware scheduler, IEEE 802.3Qbu frame preemption, and IEEE 802.1Qci ingress traffic policing.

According to the bibliography in Wilder (1976), Moore published 67 papers and one monograph, his 1932 Foundations of Point Set Theory. He is primarily remembered for his work on the foundations of topology, a topic he first touched on in his Ph.D. thesis. By the time Moore returned to the University of Texas, he had published 17 papers on point-set topology—a term he coined—including his 1915 paper "On a set of postulates which suffice to define a number-plane", giving an axiom system for plane topology. The Moore plane, Moore's road space, Moore space, and the normal Moore space conjecture are named in his honor.

If a structure has a discrete moduli (if it has no deformations, or if a deformation of a structure is isomorphic to the original structure), the structure is said to be rigid, and its study (if it is a geometric or topological structure) is topology. If it has non- trivial deformations, the structure is said to be flexible, and its study is geometry. The space of homotopy classes of maps is discrete, so studying maps up to homotopy is topology. Similarly, differentiable structures on a manifold is usually a discrete space, and hence an example of topology, but exotic R4s have continuous moduli of differentiable structures.

Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets of the larger space with the subset. For any indexed family of topological spaces, the product can be given the product topology, which is generated by the inverse images of open sets of the factors under the projection mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space.

For example, compactness and connectedness are topological properties, whereas boundedness and completeness are not. Algebraic topology is the study of topologically invariant abstract algebra constructions on topological spaces. ;Topological space: A topological space (X, T) is a set X equipped with a collection T of subsets of X satisfying the following axioms: :# The empty set and X are in T. :# The union of any collection of sets in T is also in T. :# The intersection of any pair of sets in T is also in T. :The collection T is a topology on X. ;Topological sum: See Coproduct topology. ;Topologically complete: Completely metrizable spaces (i. e.

In classical algebraic geometry (that is, the part of algebraic geometry in which one does not use schemes, which were introduced by Grothendieck around 1960), the Zariski topology is defined on algebraic varieties. The Zariski topology, defined on the points of the variety, is the topology such that the closed sets are the algebraic subsets of the variety. As the most elementary algebraic varieties are affine and projective varieties, it is useful to make this definition more explicit in both cases. We assume that we are working over a fixed, algebraically closed field k (in classical geometry k is almost always the complex numbers).

Tangles have been shown to be useful in studying DNA topology. The action of a given enzyme can be analysed with the help of tangle theory.

The gyrobifastigium topology exists in a tetragonal disphenoid with its lateral faces divided on the plane of symmetry which with specific proportions can tessellate 3-space.

Alternative hybrid architectures that share the same topology include extended cavity diode lasers and volume Bragg grating lasers, but these are not properly called DBR lasers.

The fact that large and interesting classes of non-compact spaces do in fact have compactifications of particular sorts makes compactification a common technique in topology.

For each P-block, an unrooted tree topology is calculated using RAxML. The program Quartet MaxCut is then used to calculate a supertree from these trees.

"An Early Cretaceous heterodontosaurid dinosaur with filamentous integumentary structures." Nature, 458(19): 333-336. Han et al.found a similar topology to that of Makovicky et al.

Continuity is a stronger condition: the continuity of in the natural topology (discussed below), also called multivariable continuity, which is sufficient for continuity of the composition .

In the mathematical field of low-dimensional topology, a clasper is a surface (with extra structure) in a 3-manifold on which surgery can be performed.

In mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an algebraic variety V.

The overall topology resembles a vise, with the central core of the protein at the base and the N- and C-terminal segments on the sides.

In functional analysis, a locally convex topological vector space is a bornological space if its topology can be recovered from its bornology in a natural way.

Selman Akbulut (born 1949) is a Turkish mathematician, specializing in research in topology, and geometry. He was a Professor at Michigan State University until February 2020.

In algebraic topology, a phantom map is a map between spectra such that the induced map between homology theories is zero. Phantom maps were introduced by .

In topology, a branch of mathematics, Borel's theorem, due to , says the cohomology ring of a classifying space or a classifying stack is a polynomial ring.

Sucharit Sarkar (born 1983) is an Indian topologist and associate professor of mathematics at the University of California, Los Angeles who works in low- dimensional topology.

The cladogram below follows the topology from a 2011 analysis of mitochondrial DNA sequences by Robert W. Meredith, Evon R. Hekkala, George Amato and John Gatesy.

Margherita Piazzolla Beloch (12 July 1879 in Frascati - 28 September 1976 in Rome) was an Italian mathematician who worked in algebraic geometry, algebraic topology and photogrammetry.

They may help reconcile the spin foam and canonical loop representation approaches. Recent research by Chris Duston and Matilde Marcolli introduces topology change via topspin networks.

Wacław Bolesław Marzantowicz is a Polish mathematician known for his contributions in number theory and topology. He was President of Polish Mathematical Society between 2014–2019.

Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called FK-space.

The concept of stratifolds was invented by Matthias Kreck. The basic idea is similar to that of a topologically stratified space, but adapted to differential topology.

In a similar vein, the Zariski topology on An is defined by taking the zero sets of polynomial functions as a base for the closed sets.

In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra.

He was elected a Fellow of the American Mathematical Society. Fintushel is a member of the editorial boards of Geometry & Topology and the Michigan Mathematical Journal.

Other analyses however, have found a somewhat more derived position for Preondactylus. Fossil cast The following phylogenetic analysis follows the topology of Upchurch et al. (2015).

For each topology 𝜏 on X such that (X, 𝜏) is a locally convex vector lattice, x is a quasi-interior point of its positive cone.

Constitutive criminology also has roots in chaos theory, structural coupling, strategic essentialism, topology theory, relational sets, critical race theory and intersections, autopoietic systems, and dialectical materialism.

In a cascaded topology, the propagation constant, attenuation constant and phase constant of individual sections may be simply added to find the total propagation constant etc.

In the mathematical fields of geometry and topology, a coarse structure on a set X is a collection of subsets of the cartesian product X × X with certain properties which allow the large-scale structure of metric spaces and topological spaces to be defined. The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space--such as boundedness, or the degrees of freedom of the space--do not depend on such features. Coarse geometry and coarse topology provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties.

The AMS-IX platform is continually evolving due to its rapid growth in traffic and number of connected member ports. Up until end of 2009, it is using a redundant hub- spoke architecture using a core switch and multiple edge switches. This double-star topology brings the advantage of being able to perform maintenance on the network without any impact on customer traffic, and to anticipate on fiber and equipment problems by (automatically) switching to the backup topology as soon as a failure in one of the active components occurs. The active switching topology star is determined by means of the VSRP protocol. This topology is AMS-IX version 3. However, since 2009; AMS-IX platform has migrated from a pure Layer2 network to a VPLS/MPLS network (using Brocade hardware) in order to cope with future growth (this is AMS-IX version 4).

As was mentioned above, computing the fundamental group of even relatively simple topological spaces tends to be not entirely trivial, but requires some methods of algebraic topology.

Marshall Harvey Stone (April 8, 1903 – January 9, 1989) was an American mathematician who contributed to real analysis, functional analysis, topology and the study of Boolean algebras.

In algebraic topology, an area of mathematics, a homeotopy group of a topological space is a homotopy group of the group of self-homeomorphisms of that space.

Work of Nikolai Luzin and Richard Laver shows that this conjecture is independent of the ZFC axioms. This article is about the Borel conjecture in geometric topology.

If installed correctly, it can maintain reference ground potential much better than a star topology in a similar application across a wider range of frequencies and currents.

John Willard Morgan (born March 21, 1946) is an American mathematician, with contributions to topology and geometry. He is, as of 2020, Professor Emeritus at Columbia University.

Carl Barnett Allendoerfer (April 4, 1911 - September 29, 1974) was an American mathematician in the mid-twentieth century, known for his work in topology and mathematics education.

Burt James Totaro, FRS (b. 1967), is an American mathematician, currently a Professor at the University of California, Los Angeles, specializing in algebraic geometry and algebraic topology.

Steinberg's basis was again motivated by a problem on the topology of homogeneous spaces; the basis arises in describing the T-equivariant K-theory of K/T.

As with topological groups, some authors require the topology to be Hausdorff.A. V. Arhangelskii. Topological spaces connected to algebraic structures Compact paratopological groups are automatically topological groups.

Note that has the following properties: #It is subadditive: . #It is homogeneous: for all scalars . #It is nonnegative: . Therefore, is a seminorm on , with an induced topology.

In the mathematical field of geometric topology, a Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.

Richard James Milgram (born 5 December 1939 in South Bend, Indiana) is an American mathematician, specializing in algebraic topology. He is the son of mathematician Arthur Milgram.

Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable, weakly analytic, etc.) if they are continuous (respectively, differentiable, analytic, etc.) with respect to the weak topology.

If M is an m-manifold and N is an n-manifold, the Cartesian product M×N is a (m+n)-manifold when given the product topology.

The circle of center 0 and radius 1 in the complex plane is a compact Lie group with complex multiplication. In mathematics, a compact (topological) group is a topological group whose topology is compact. Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well- understood theory, in relation to group actions and representation theory.

MultiCluster-1 series were statically configurable systems and could be tailored to specific user requirements such as number of processors, amount of memory, and I/0 configuration, as weil as system topology. The required processor topology could be configured by using UniLink connection; fed through the special back plane. In addition, four external sockets were provided. Multicluster-2 used network configuration units (NCUs) that provided flexible, dynamically configurable interconnection networks.

Nesbitt et al. named this group the Aphanosauria, defined as the most inclusive clade containing Teleocrater rhadinus and Yarasuchus deccanensis but not Passer domesticus or Crocodylus niloticus. The results of the analyses are reproduced below, based primarily on the Ezcurra dataset but incorporating the avemetatarsalian topology of the Nesbitt dataset. The inclusion of Scleromochlus altered the topology obtained to varying extents, although both analyses recovered it as an avemetatarsalian.

A topological space X is said to be disconnected if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice.

In point-set topology, Kuratowski's closure-complement problem asks for the largest number of distinct sets obtainable by repeatedly applying the set operations of closure and complement to a given starting subset of a topological space. The answer is 14. This result was first published by Kazimierz Kuratowski in 1922. The problem gained wide exposure three decades later as an exercise in John L. Kelley's classic textbook General Topology.

In the analysis of Andres and colleagues, Ludodactylus is classified just outside Ornithocheiridae and Anhangueridae as a derived member of the more inclusive group Anhangueria. In 2018, a topology by Nicholas Longrich and colleagues had recovered Ludodactylus in a different position, though still within Anhangueria. They placed Ludodactylus as the sister taxon of Guidraco instead, and also in a more basal position. Topology 1: Andres et al. (2014).

Apart from using h-principle to study the flexibility of local geometric models, Murphy's work uses cut-and-paste/surgery techniques from smooth topology. She also works on exploring the interaction of symplectic/contact topology with geometric invariants, such as those coming from pseudo-holomorphic curves or constructible sheaves. Murphy received the grants from National Science Foundation for the period 2019-2022 on the topic "Flexible Stein Manifolds and Fukaya Categories".

Mary Wynne Warner (22 June 1932 – 1 April 1998) was a Welsh mathematician, specializing in fuzzy mathematics.M. W. Warner, "Fuzzy topology with respect to continuous lattices," Fuzzy Sets and Systems 35(1)(1990): 85–91. doi:10.1016/0165-0114(90)90020-7M. W. Warner, "Towards a Mathematical Theory of Fuzzy Topology" in R. Lowen and M. R. Roubens, eds., Fuzzy Logic: State of the Art (Springer 1993): 83–94.

Euler's Gem: The Polyhedron Formula and the Birth of Topology is a book on the formula V-E+F=2 for the Euler characteristic of convex polyhedra and its connections to the history of topology. It was written by David Richeson and published in 2008 by the Princeton University Press, with a paperback edition in 2012. It won the 2010 Euler Book Prize of the Mathematical Association of America.

For a network with three branches there are four possible topologies; Figure 1.4. Series and parallel topologies with three branches Note that the parallel-series topology is another representation of the Delta topology discussed later. Series and parallel topologies can continue to be constructed with greater and greater numbers of branches ad infinitum. The number of unique topologies that can be obtained from n branches is 2n-1.

Graph theory is the branch of mathematics dealing with graphs. In network analysis, graphs are used extensively to represent a network being analysed. The graph of a network captures only certain aspects of a network; those aspects related to its connectivity, or, in other words, its topology. This can be a useful representation and generalisation of a network because many network equations are invariant across networks with the same topology.

The various services are encoded, modulated and upconverted onto RF carriers, combined onto a single electrical signal and inserted into a broadband optical transmitter. This optical transmitter converts the electrical signal to a downstream optically modulated signal that is sent to the nodes. Fiber optic cables connect the headend or hub to optical nodes in a point-to-point or star topology, or in some cases, in a protected ring topology.

4, 374–382. with Allyn Jackson of the American Mathematical Society, he explained that he sees "networks as discrete versions of harmonic theory", so his experience with network synthesis and electronic filter topology introduced him to algebraic topology. Bott met Arnold S. Shapiro at the IAS and they worked together. He studied the homotopy theory of Lie groups, using methods from Morse theory, leading to the Bott periodicity theorem (1957).

OVN is a network virtualization platform that separates the physical network topology from the logical one. Users are able to connect virtual and physical interfaces with logical switches and routers, regardless of the underlying physical topology. Users are also able to define security policies and load-balancing to these logical instances. OVN uses Open vSwitch for its switching fabric and uses tunnels to provide the logical/physical separation.

Fundamenta Mathematicae is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical systems. Originally it only covered topology, set theory, and foundations of mathematics: it was the first specialized journal in the field of mathematics..... It is published by the Mathematics Institute of the Polish Academy of Sciences.

Edwin Henry Spanier (August 8, 1921 - October 11, 1996) was an American mathematician at the University of California at Berkeley, working in algebraic topology. He co-invented Spanier–Whitehead duality and Alexander–Spanier cohomology, and wrote what was for a long time the standard textbook on algebraic topology . Spanier attended the University of Minnesota, graduating in 1941. During World War II, he served in the United States Army Signal Corps.

A topological group is a group together with a topology with respect to which the group composition and the inversion are continuous. Such a group is called compact, if any cover of G, which is open in the topology, has a finite subcover. Closed subgroups of a compact group are compact again. Let G be a compact group and let V be a finite-dimensional \Complex–vector space.

This result is often implicitly used to extend affine geometry to projective geometry: it is entirely typical for an affine variety to acquire singular points on the hyperplane at infinity, when its closure in projective space is taken. Resolution says that such singularities can be handled rather as a (complicated) sort of compactification, ending up with a compact manifold (for the strong topology, rather than the Zariski topology, that is).

UC frames may only enter a Sercos III network through a Sercos III- compliant port. This can be achieved two different ways. One is to employ the unused Sercos III port at the end of a Sercos III network configured in line topology, as shown to the right. In a network configured in ring topology, the ring can be temporarily broken at any point to also attach a device.

It represents the view that most botanists had held up to that time. It was supported by morphological studies, but never received strong support in molecular phylogenetic studies. In 2015, a phylogenetic study showed strong statistical support for the following topology of the orchid tree, using 9 kb of plastid and nuclear DNA from 7 genes, a topology that was confirmed by a phylogenomic study in the same year.

Metriorhynchoidea is a stem-based taxon defined in 2009 as the most inclusive clade consisting of Metriorhynchus geoffroyii, but not Teleosaurus cadomensis. The cladogram below follows the topology from a 2011 analysis by Andrea Cau and Federico Fanti. Note that the same topology was obtained in Mark T. Young and Marco Brandalise de Andrade, 2009 and Mark T. Young, Stephen L. Brusatte, Marcello Ruta and Marco Brandalise de Andrade, 2010.

Torus fusion (tofu) is a proprietary computer network topology for supercomputers developed by Fujitsu. It is a variant of the torus interconnect. The system has been used in the K computer and the Fugaku supercomputer (and their derivatives). Tofu has a six-dimensional mesh/torus topology, a scalability of over 100,000 nodes, and full-duplex links that have a peak bandwidth of 10 GB/s (5 GB/s per direction).

More generally one is interested in properties and invariants of smooth manifolds that are carried over by diffeomorphisms, another special kind of smooth mapping. Morse theory is another branch of differential topology, in which topological information about a manifold is deduced from changes in the rank of the Jacobian of a function. For a list of differential topology topics, see the following reference: List of differential geometry topics.

The crystal structures of a number of Rieske proteins are known. The overall fold, comprising two subdomains, is dominated by antiparallel β-structure and contains variable numbers of α-helices. The smaller "cluster-binding" subdomains in mitochondrial and chloroplast proteins are virtually identical, whereas the large subdomains are substantially different in spite of a common folding topology. The [Fe2S2] cluster-binding subdomains have the topology of an incomplete antiparallel β-barrel.

Though the details of the aerodynamics depend very much on the topology, some fundamental concepts apply to all turbines. Every topology has a maximum power for a given flow, and some topologies are better than others. The method used to extract power has a strong influence on this. In general, all turbines may be grouped as being either lift-based, or drag-based; the former being more efficient.

The park is located in the subdivision of the Sierra Madre Occidental known as Sierra Tarahumara, near the municipality of Ocampo, Chihuahua. The dramatic topology was created by deep tectonic plate movements causing large fractures and rising rifts. The violent movement of the terrain resulted in deep canyons and high mountains. The present-day topology has been changed over thousands of years by wind erosion and the Basaseachic River.

In geometric topology, a cellular decomposition G of a manifold M is a decomposition of M as the disjoint union of cells (spaces homeomorphic to n-balls Bn). The quotient space M/G has points that correspond to the cells of the decomposition. There is a natural map from M to M/G, which is given the quotient topology. A fundamental question is whether M is homeomorphic to M/G.

Partially connected mesh topology In a partially connected network, certain nodes are connected to exactly one other node; but some nodes are connected to two or more other nodes with a point-to-point link. This makes it possible to make use of some of the redundancy of mesh topology that is physically fully connected, without the expense and complexity required for a connection between every node in the network.

In algebraic topology, a branch of mathematics, the Čech-to-derived functor spectral sequence is a spectral sequence that relates Čech cohomology of a sheaf and sheaf cohomology.

For infinite-dimensional associative division algebras, the most important cases are those where the space has some reasonable topology. See for example normed division algebras and Banach algebras.

Valentin Alexandre Poénaru (born 1932 in Bucharest) is a Romanian–French mathematician. He was a Professor of Mathematics at University of Paris-Sud, specializing in low-dimensional topology.

The history of the separation axioms in general topology has been convoluted, with many meanings competing for the same terms and many terms competing for the same concept.

Julia Elizabeth Bergner is a mathematician specializing in algebraic topology, homotopy theory, and higher category theory. She is an associate professor of mathematics at the University of Virginia.

Journal of Vertebrate Paleontology 22 (1): 91–103. The cladogram below follows the most resolved topology from a 2011 analysis by paleontologists Takuya Konishi and Michael W. Caldwell.

In number theory, more specifically in p-adic analysis, Krasner's lemma is a basic result relating the topology of a complete non-archimedean field to its algebraic extensions.

CI Deployment Topology. Graphic created by J.B. Matthews. The Cyberinfrastructure component links marine infrastructure to scientists and users. It manages and integrates data from the different OOI sensors.

At the cellular level podocalyxin has also been shown to regulate the size and topology of apical cell domains and act as a potent inducer of microvillus formation.

A consequence is that any uniform structure can be defined as above by a (possibly uncountable) family of pseudometrics (see Bourbaki: General Topology Chapter IX §1 no. 4).

Shelly Lynn Harvey is a professor of Mathematics at Rice University. Her research interests include knot theory, low-dimensional topology, and group theory.Curriculum vitae, retrieved 2014-12-21.

The Hopf theorem (named after Heinz Hopf) is a statement in differential topology, saying that the topological degree is the only homotopy invariant of continuous maps to spheres.

A JW algebra is a Jordan subalgebra of the Jordan algebra of self-adjoint operators on a complex Hilbert space that is closed in the weak operator topology.

In 2018 Schmidt and Michael Winter published Relational Topology which reviews classical mathematical structures, such as binary operations and topological space, through the lens of calculus of relations.

In mathematics, especially in algebraic topology, the homotopy limit and colimit are variants of the notions of limit and colimit. They are denoted by holim and hocolim, respectively.

Proceedings of the American Mathematical Society publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics.

Gérard Debreu The Theory of Value: An axiomatic analysis of economic equilibrium, 1959 In the 1960s and 1970s, however, Gérard Debreu and Stephen Smale led a revival of the use of differential calculus in mathematical economics. In particular, they were able to prove the existence of a general equilibrium, where earlier writers had failed, through the use of their novel mathematics: Baire category from general topology and Sard's lemma from differential topology and differential geometry. Their publications initiated a period of research "characterized by the use of elementary differential topology": "almost every area in economic theory where the differential approach has been pursued, including general equilibrium" was covered by Mas-Colell's monograph on differentiable analysis and economics. Mas-Colell's book "offers a synthetic and thorough account of a major recent development in general equilibrium analysis, namely, the largely successful reconstruction of the theory using modern ideas of differential topology", according to its back cover.

If each of the bonding maps f_{i}^{j} is an embedding of TVSs onto proper vector subspaces and if the system is directed by with its natural ordering, then the resulting limit is called a strict (countable) direct limit. In such a situation we may assume without loss of generality that each is a vector subspace of and that the subspace topology induced on by is identical to the original topology on . In the category of locally convex topological vector spaces, the topology on a strict inductive limit of Fréchet spaces can be described by specifying that an absolutely convex subset is a neighborhood of if and only if is an absolutely convex neighborhood of in for every .

The majority of modern commercial Hi-fi amplifier designs have until recently used class-AB topology (with more or less pure low-level class-A capability depending on the standing bias current used), in order to deliver greater power and efficiency, typically 12–25 watts and higher. Contemporary designs normally include at least some negative feedback. However, class-D topology (which is vastly more efficient than class B) is more and more frequently applied where traditional design would use class AB because of its advantages in both weight and efficiency. Class-AB push–pull topology is nearly universally used in tube amps for electric guitar applications that produce power of more than about 10 watts.

Filters designed using Cauer's topology of the first form are low-pass filters consisting of a ladder network of series inductors and shunt capacitors. A low-pass filter implemented in Cauer topology Attaching generators to the input and output ports Nodes of the dual network Components of the dual network The dual network with the original removed and slightly redrawn to make the topology clearer It can now be seen that the dual of a Cauer low-pass filter is still a Cauer low-pass filter. It does not transform into a high-pass filter as might have been expected. Note, however, that the first element is now a shunt component instead of a series component.

Ashoke Sen has conjectured that, in the absence of a topologically nontrivial NS 3-form flux, all IIB brane configurations can be obtained from stacks of spacefilling D9 and anti D9 branes via tachyon condensation. The topology of the resulting branes is encoded in the topology of the gauge bundle on the stack of the spacefilling branes. The topology of the gauge bundle of a stack of D9s and anti D9s can be decomposed into a gauge bundle on the D9's and another bundle on the anti D9's. Tachyon condensation transforms such a pair of bundles to another pair in which the same bundle is direct summed with each component in the pair.

Koetsier and van Mill write that many of these younger topologists experienced compactification at first hand while trying to squeeze into the back seat of De Groot's small Mercedes. McDowell. writes, "His students essentially constitute the topology faculties at the Dutch universities." The deep influence of de Groot on Dutch topology may be seen in the complex academic genealogy of his namesake Johannes Antonius Marie de Groot (shown in the illustration): the later de Groot, a 1990 Ph.D. in topology, is the senior de Groot's academic grandchild, great-grandchild, and great-great-grandchild via four different paths of academic supervision.. De Groot was elected in 1969 to the Royal Dutch Academy of Sciences.

In topology, a quotient space is a topological space formed on the set of equivalence classes of an equivalence relation on a topological space, using the original space's topology to create the topology on the set of equivalence classes. In abstract algebra, congruence relations on the underlying set of an algebra allow the algebra to induce an algebra on the equivalence classes of the relation, called a quotient algebra. In linear algebra, a quotient space is a vector space formed by taking a quotient group, where the quotient homomorphism is a linear map. By extension, in abstract algebra, the term quotient space may be used for quotient modules, quotient rings, quotient groups, or any quotient algebra.

Then S is not a base for any topology on R. To show this, suppose it were. Then, for example, (−∞, 1) and (0, ∞) would be in the topology generated by S, being unions of a single base element, and so their intersection (0,1) would be as well. But (0, 1) clearly cannot be written as a union of elements of S. Using the alternate definition, the second property fails, since no base element can "fit" inside this intersection. Given a base for a topology, in order to prove convergence of a net or sequence it is sufficient to prove that it is eventually in every set in the base which contains the putative limit.

Residual topology is a descriptive stereochemical term to classify a number of intertwined and interlocked molecules, which cannot be disentangled in an experiment without breaking of covalent bonds, while the strict rules of mathematical topology allow such a disentanglement. Examples of such molecules are rotaxanes, catenanes with covalently linked rings (so-called pretzelanes), and open knots (pseudoknots) which are abundant in proteins. The term "residual topology" was suggested on account of a striking similarity of these compounds to the well-established topologically nontrivial species, such as catenanes and knotanes (molecular knots). The idea of residual topological isomerism introduces a handy scheme of modifying the molecular graphs and generalizes former efforts of systemization of mechanically bound and bridged molecules.

Since topological indistinguishability is an equivalence relation on any topological space X, we can form the quotient space KX = X/≡. The space KX is called the Kolmogorov quotient or T0 identification of X. The space KX is, in fact, T0 (i.e. all points are topologically distinguishable). Moreover, by the characteristic property of the quotient map any continuous map f : X → Y from X to a T0 space factors through the quotient map q : X → KX. Although the quotient map q is generally not a homeomorphism (since it is not generally injective), it does induce a bijection between the topology on X and the topology on KX. Intuitively, the Kolmogorov quotient does not alter the topology of a space.

For a chain with two binary contacts, three arrangements are available: parallel, series and crossed. For a chain with n contacts, the topology can be described by an n by n matrix in which each element illustrates the relation between a pair of contacts and may take one of the three states, P, S and X. Multivalent contacts can also be categorised in full or via decomposition into several binary contacts. Similarly, circuit topology allows for classification of the pairwise arrangements of chain crossings and tangles, thus providing a complete 3D description of folded chains. Circuit topology has implications for folding kinetics and molecular evolution and has been applied to engineer polymers including protein origami.

The topology on that will now be defined, which is in general different from the product topology, allows for a characterization of convergence of nets in in terms of convergence of the graphs (which are sets) of maps in . :Let denote the _topology for convergence of graphs_ on that is generated by the subbasis consisting of all sets of the form :: :where ranges over and ranges over the open subset of . If is a topological space and if each is closed in then is weaker than the product topology on . More importantly, if and if is a net in , then in if and only if its net of graphs converges to in .

A quad-edge data structure is a computer representation of the topology of a two-dimensional or three-dimensional map, that is, a graph drawn on a (closed) surface.

In algebraic topology, the Hattori–Stong theorem, proved by and , gives an isomorphism between the stable homotopy of a Thom spectrum and the primitive elements of its K-homology.

This viewpoint makes it easier to be precise about such concepts as covering maps. The generalization to a locally partially ordered space is studied in ; see also Directed topology.

If each Xi is homeomorphic to a fixed space A, then the disjoint union X is homeomorphic to the product space A × I where I has the discrete topology.

Laura J. Person is an American mathematician specializing in low-dimensional topology. She is a Distinguished Teaching Professor of Mathematics at the State University of New York at Potsdam.

In mathematics, topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology.

Tsavdaridis, Konstantinos Daniel; Kingman, James; Toropov, Vassilli (31 July 2014). "Application of structural topology optimisation to perforated steel beams". Computers and Structures. 158: 108–123. doi:10.1016/j.compstruc.2015.05.004.

In 1993, he and his co-author Lowell E. Jones introduced the Farrell–Jones conjecture. and made contributions on it. The conjecture plays an important role in manifold topology.

In mathematics, a multiplicative sequence or m-sequence is a sequence of polynomials associated with a formal group structure. They have application in the cobordism ring in algebraic topology.

In geometric topology, the spherical space form conjecture states that a finite group acting on the 3-sphere is conjugate to a group of isometries of the 3-sphere.

The shapefile format does not have the ability to store topological information. The ESRI ArcInfo coverages and personal/file/enterprise geodatabases do have the ability to store feature topology.

He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to Floer homology and low-dimensional topology and service to the mathematical community".

Judith "Judy" Roitman (born November 12, 1945) is a mathematician, a retired professor at the University of Kansas. She specializes in set theory, topology, Boolean algebras, and mathematics education.

The architecture is designed to scale almost indefinitely, with 4 e-links allowing multiple chips to be combined in a grid topology, allowing for systems with thousands of cores.

In differential topology, a critical value of a differentiable function between differentiable manifolds is the image (value of) ƒ(x) in N of a critical point x in M.

Wolfhart Zimmermann (17 February 1928 – 18 September 2016) was a German theoretical physicist. Zimmermann attained a doctorate in 1950 at Freiburg im Breisgau in topology ("Eine Kohomologietheorie topologischer Räume").

This page lists some properties of sets of real numbers. The general study of these concepts forms descriptive set theory, which has a rather different emphasis from general topology.

Jennifer Carol Schultens (born 1965) is an American mathematician specializing in low-dimensional topology and knot theory. She is a professor of mathematics at the University of California, Davis.

The NicO family within the LysE superfamily may have a common origin with the TOG superfamily, having lost TMSs 1 and 4 in the 8 TMS TOG superfamily topology.

In mathematics, in the realm of topology, a paranormal space is a topological space in which every countable discrete collection of closed sets has a locally finite open expansion.

In: Viro O.Y., Vershik A.M. (eds) Topology and Geometry — Rohlin Seminar. Lecture Notes in Mathematics, vol 1346. Springer, Berlin, HeidelbergBarden, D. "Simply Connected Five-Manifolds." Annals of Mathematics, vol.

In mathematics, a Borel equivalence relation on a Polish space X is an equivalence relation on X that is a Borel subset of X × X (in the product topology).

In geometric topology, a branch of mathematics, the Bing shrinking criterion, introduced by , is a method for showing that a quotient of a space is homeomorphic to the space.

Michel André (26 March 1936 – 9 July 2009) was a Swiss mathematician, specializing in non-commutative algebra and its applications to topology. He is known for André–Quillen cohomology.

In topology and analysis, filters are used to define convergence in a manner similar to the role of sequences in a metric space. In topology and related areas of mathematics, a filter is a generalization of a net. Both nets and filters provide very general contexts to unify the various notions of limit to arbitrary topological spaces. A sequence is usually indexed by the natural numbers, which are a totally ordered set.

Jean Cerf studied at the École Normale Supérieure, graduating in sciences in 1947. After passing his agrégation in mathematics in 1950, he obtained a doctorate with thesis supervised by Henri Cartan. Cerf became a maître de conférences at the University of Lille and was later appointed a professor at the University of Paris XI. He was also a director of research at CNRS.Cerf, Biographie Cerf's research deals with differential topology, cobordism, and symplectic topology.

If F is given a new metric without changing the topology, this metric can be extended to the entire space without changing the topology. The work Gestufte Räume appeared in 1935. Here Hausdorff discussed spaces which fulfilled the Kuratowski closure axioms up to just the axiom of idempotence. He named them graded spaces (often also called closure spaces) and used them in the study of the relationships between the Fréchet limit spaces and topological spaces.

It also includes the study of videogames, graphic novels, the infinite canvas, and narrative sculptures linked to topology and graph theory. Félix Lambert, 2015, "Narrative sculptures: graph theory, topology and new perspectives in narratology." However, constituent analysis of a type where narremes are considered to be the basic units of narrative structure could fall within the areas of linguistics, semiotics, or literary theory.Henri Wittmann, "Théorie des narrèmes et algorithmes narratifs," Poetics 4.1 (1975): 19–28.

Campbell's sketch of the low-pass version of his filter from his 1915 patentGeorge A, Campbell, Electric wave-filter, , filed 15 July 1915, issued 22 May 1917. showing the now ubiquitous ladder topology with capacitors for the ladder rungs and inductors for the stiles. Filters of more modern design also often adopt the same ladder topology as used by Campbell. It should be understood that although superficially similar, they are really quite different.

John von Neumann's work on functional analysis and topology broke new ground in mathematics and economic theory.Neumann, John von, and Oskar Morgenstern (1944) Theory of Games and Economic Behavior, Princeton. It also left advanced mathematical economics with fewer applications of differential calculus. In particular, general equilibrium theorists used general topology, convex geometry, and optimization theory more than differential calculus, because the approach of differential calculus had failed to establish the existence of an equilibrium.

Weeks is particularly interested in using topology to understand the spatial universe.. His book The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds (Marcel Dekker, 1985, ) explores the geometry and topology of low- dimensional manifolds.Review by Thomas Banchoff (1987), Mathematical Reviews, .Review by Alan H. Durfee (1989), American Mathematical Monthly 96 (7): 660–662, . The second edition (2002, ) explains some of his work in applying the material to cosmology.

Let M be a manifold. M has a category of open sets O(M) because it is a topological space, and it gets a topology as in the above example. For two open sets U and V of M, the fiber product U ×M V is the open set U ∩ V, which is still in O(M). This means that the topology on O(M) is defined by a pretopology, the same pretopology as before.

The topology of any given 7-polytope is defined by its Betti numbers and torsion coefficients.Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008. The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.

The topology of any given 9-polytope is defined by its Betti numbers and torsion coefficients.Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008. The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.

The topology of any given 5-polytope is defined by its Betti numbers and torsion coefficients.Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008. The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.

Inverse topology micellar cubic phases (such as the Fd3m phase) are observed for some type II amphiphiles at very high amphiphile concentrations. These aggregates, in which water is the minority phase, have a polar-apolar interface with a negative mean curvature. The structures of the normal topology micellar cubic phases that are formed by some types of amphiphiles (e.g. the oligoethyleneoxide monoalkyl ether series of non-ionic surfactants are the subject of debate.

Intermediate nodes not only boost the signal, but cooperatively pass data from point A to point B by making forwarding decisions based on their knowledge of the network, i.e. perform routing by first deriving the topology of the network. Wireless mesh networks is a relatively "stable-topology" network except for the occasional failure of nodes or addition of new nodes. The path of traffic, being aggregated from a large number of end users, changes infrequently.

Graphs have path connected subsets, namely those subsets for which every pair of points has a path of edges joining them. But it is not always possible to find a topology on the set of points which induces the same connected sets. The 5-cycle graph (and any n-cycle with n > 3 odd) is one such example. As a consequence, a notion of connectedness can be formulated independently of the topology on a space.

NCA does the required analyses and provides display of the feed point of various network loads. Based on the status of all the switching devices such as circuit breaker (CB), Ring Main Unit (RMU) and/or isolators that affect the topology of the network modeled, the prevailing network topology is determined. The NCA further assists the operator to know operating state of the distribution network indicating radial mode, loops and parallels in the network.

Most commonly the cause is a switching loop in the Ethernet wiring topology (i.e. two or more paths exist between end stations). As broadcasts and multicasts are forwarded by switches out of every port, the switch or switches will repeatedly rebroadcast broadcast messages and flood the network. Since the Layer 2 header does not support a time to live (TTL) value, if a frame is sent into a looped topology, it can loop forever.

In mathematical complex analysis, Radó's theorem, proved by , states that every connected Riemann surface is second-countable (has a countable base for its topology). The Prüfer surface is an example of a surface with no countable base for the topology, so cannot have the structure of a Riemann surface. The obvious analogue of Radó's theorem in higher dimensions is false: there are 2-dimensional connected complex manifolds that are not second-countable.

In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, or codiscrete. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space is a pseudometric space in which the distance between any two points is zero.

Ehresmann first investigated the topology and homology of manifolds associated with classical Lie groups, such as Grassmann manifolds and other homogeneous spaces. He developed the concept of fiber bundle, building on work by Herbert Seifert and Hassler Whitney. Norman Steenrod was working in the same direction in the USA, but Ehresmann was particularly interested in differentiable (smooth) fiber bundles, and in differential-geometric aspects of these. He was a pioneer of differential topology.

Systems with a known topology can be initialized in a system specific manner without affecting interoperability. The RapidIO system initialization specification supports system initialization when system topology is unknown or dynamic. System initialization algorithms support the presence of redundant hosts, so system initialization need not have a single point of failure. Each system host recursively enumerates the RapidIO fabric, seizing ownership of devices, allocating device IDs to endpoints and updating switch routing tables.

As a post- doctoral student, he was at the Mathematical Sciences Research Institute (MSRI) in Berkeley, and then a tutor at Jesus College, Oxford. From 1998 until shortly before his death he was a professor at the Pennsylvania State University. His research interests center around index theorems, coarse geometry, operator algebras, noncommutative geometry, and the Novikov conjecture in differential topology. He was an editor of the Journal of Noncommutative Geometry and the Journal of Topology.

Leray lectured mainly on calculus and topology, concealing his expertise in fluid dynamics and mechanics since he feared being forced to work on German military projects. He also studied algebraic topology, publishing several papers after the war on spectral sequences and sheaf theory. Other notable figures of the University were the embryologist Étienne Wolff and the geologist François Ellenberger. The syllabus also included such subjects as law, biology, psychology, Arabic, music, moral theology, and astronomy.

Graded Hopf algebras are often used in algebraic topology: they are the natural algebraic structure on the direct sum of all homology or cohomology groups of an H-space. Locally compact quantum groups generalize Hopf algebras and carry a topology. The algebra of all continuous functions on a Lie group is a locally compact quantum group. Quasi-Hopf algebras are generalizations of Hopf algebras, where coassociativity only holds up to a twist.

With the exception of the yeast protein (627 amino acyl residues), all characterized members of the family are of 256-285 residues in length and exhibit 6-8 putative transmembrane α-helical spanners (TMSs). In one case, that of the E. coli FocA (TC# 1.A.16.1.1) protein, a 6 TMS topology has been established. The yeast protein has a similar apparent topology but has a large C-terminal hydrophilic extension of about 400 residues.

Kocinac has published over 160 papers and four books in topology, real analysis and fields of sets. He has actively promoted research on selection principles, as a fruitful collaborator and as an organizer of the first conferences in a series of international workshops titled Coverings, Selections and Games in Topology. The fourth of this series, held in Caserta, Italy, in June 2012 was dedicated to him on the occasion of his sixty-fifth birthday.

In the same year however, a study by Pêgas et al. placed Camposipterus within the clade Targaryendraconia, and specifically within the family Cimoliopteridae as the sister taxon of both Aetodactylus and Cimoliopterus:Rodrigo V. Pêgas, Borja Holgado & Maria Eduarda C. Leal (2019) On Targaryendraco wiedenrothi gen. nov. (Pterodactyloidea, Pteranodontoidea, Lanceodontia) and recognition of a new cosmopolitan lineage of Cretaceous toothed pterodactyloids, Historical Biology, Topology 1: Jacobs et al. (2019). Topology 2: Pêgas et al. (2019).

All manufactured components have finite size and well behaved boundaries, so initially the focus was on mathematically modeling rigid parts made of homogeneous isotropic material that could be added or removed. These postulated properties can be translated into properties of subsets of three- dimensional Euclidean space. The two common approaches to define solidity rely on point-set topology and algebraic topology respectively. Both models specify how solids can be built from simple pieces or cells.

Sierpiński authored 724 papers and 50 books, mostly in Polish. His book Cardinal and Ordinal Numbers was originally published in English in 1958. Two books, Introduction to General Topology (1934) and General Topology (1952) were translated into English by Canadian mathematician Cecilia Krieger. Another book, Pythagorean Triangles (1954), was translated into English by Indian mathematician Ambikeshwar Sharma, published in 1962, and republished by Dover Books in 2003; it also has a Russian translation.

The topological classification of Calabi-Yau manifolds has important implications in string theory, as different manifolds can sustain different kinds of strings.Yau, S. & Nadis, S.; The Shape of Inner Space, Basic Books, 2010. In cosmology, topology can be used to describe the overall shape of the universe.The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds 2nd ed (Marcel Dekker, 1985, ) This area of research is commonly known as spacetime topology.

In the case of a Euclidean space, both topological dimensions are equal to n. Every subset of a topological space is itself a topological space (in contrast, only linear subsets of a linear space are linear spaces). Arbitrary topological spaces, investigated by general topology (called also point-set topology) are too diverse for a complete classification up to homeomorphism. Compact topological spaces are an important class of topological spaces ("species" of this "type").

Let X = {a,b} be a set with 2 elements. There are four distinct topologies on X: #{∅, {a,b}} (the trivial topology) #{∅, {a}, {a,b}} #{∅, {b}, {a,b}} #{∅, {a}, {b}, {a,b}} (the discrete topology) The second and third topologies above are easily seen to be homeomorphic. The function from X to itself which swaps a and b is a homeomorphism. A topological space homeomorphic to one of these is called a Sierpiński space.

Frequently the Fréchet spaces that arise in practical applications of the derivative enjoy an additional property: they are tame. Roughly speaking, a tame Fréchet space is one which is almost a Banach space. On tame spaces, it is possible to define a preferred class of mappings, known as tame maps. On the category of tame spaces under tame maps, the underlying topology is strong enough to support a fully fledged theory of differential topology.

The cladogram below follows the topology from a 2011 analysis of mitochondrial DNA sequences by Robert W. Meredith, Evon R. Hekkala, George Amato and John Gatesy. The cladogram below follows the topology from a 2012 analysis of morphological traits by Christopher A. Brochu and Glenn W. Storrs. Many extinct species of Crocodylus might represent different genera. C. suchus was not included, because its morphological codings were identical to these of C. niloticus.

Below is a cladogram showing the phylogenetic placement of Azhdarcho within the clade Neoazhdarchia. The cladogram is based on a topology recovered by Brian aAndres and Timothy Myers in 2013.

Cantor's intersection theorem refers to two closely related theorems in general topology and real analysis, named after Georg Cantor, about intersections of decreasing nested sequences of non-empty compact sets.

Call this complicated space K. A branched surface is a space that is locally modeled on K.Li, Tao. "Laminar Branched Surfaces in 3-manifolds." Geometry and Topology 6.153 (2002): 194.

Richard Schoen and Shing Tung Yau. Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature. Comment. Math. Helv. 51 (1976), no. 3, 333–341.

In terms of the natural topology on the fundamental group, a locally path-connected space is semi-locally simply connected if and only if its quasitopological fundamental group is discrete.

A torus or a cylinder can both be given flat metrics, but differ in their topology. Other topologies are also possible for curved space. See also shape of the universe.

Below is a cladogram following a topology by Pêgas et al. (2019). In the analyses, they recovered Targaryendraconia as the sister taxon of Anhangueria within the more inclusive group Ornithocheirae.

Julia Elisenda (Eli) Grigsby is an American mathematician who works as a professor at Boston College. Her research concerns low-dimensional topology, including knot theory and category-theoretic knot invariants.

Any Cauchy space is also a convergence space, where a filter F converges to x if F ∩ U(x) is Cauchy. In particular, a Cauchy space carries a natural topology.

In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology.

José Adem (born in Tuxpan, Veracruz, October 27, 1921; died February 14, 1991) was a Mexican mathematician who worked in algebraic topology, and proved the Adem relations between Steenrod squares.

Georges Henri Reeb (12 November 1920 – 6 November 1993) was a French mathematician. He worked in differential topology, differential geometry, differential equations, topological dynamical systems theory and non-standard analysis.

His primary research interest is algebraic topology; his best-cited workGoogle scholar, accessed 2010-01-23. consists of two papers in the Annals of Mathematics on "nilpotence and stable homotopy".

Hans Hahn (; 27 September 1879 – 24 July 1934) was an Austrian mathematician who made contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory.

DSL Rings (DSLR) or Bonded DSL Rings is a ring topology that uses DSL technology over existing copper telephone wires to provide data rates of up to 400 Mbit/s.

In 2009, Topology won the Outstanding Contribution by an Organisation award at the Australasian Performing Right Association (APRA) Classical Music Awards for their work on the 2008 Brisbane Powerhouse Series.

If F is equipped with the order topology arising from the total order ≤, then the axioms guarantee that the operations + and × are continuous, so that F is a topological field.

A torus bundle, in the sub-field of geometric topology in mathematics, is a kind of surface bundle over the circle, which in turn is a class of three- manifolds.

As of version 6.0 (2012), a personal firewall solution and Network Topology analysis software was introduced into the product along with a packet tracer and analyzer for further administrative use.

In particular we find that vector bundles inducing homotopic maps to the Grassmannian are isomorphic. Here the definition of homotopic relies on a notion of continuity, and hence a topology.

Below is a cladogram showing the phylogenetic placement of Arambourgiania within the clade Neoazhdarchia. The cladogram is based on a topology recovered by Brian Andres and Timothy Myers in 2013.

For this reason Ethersound is best used in applications suitable to a daisy chain network topology or in live sound applications that benefit from its low point-to-point latency.

In topology, a branch of mathematics, a cubical set is a set-valued contravariant functor on the category of (various) n-cubes. See the references for the more precise definitions.

Experimental Mathematics was established in 1992 by David Epstein, Silvio Levy, and Klaus Peters.Foreword by Igor Rivin, Colin Rourke and Caroline Series. Epstein birthday schrift. Geometry & Topology Monographs, vol. 1.

In comparison, Controller Area Networks, common in vehicles, are primarily distributed control system networks of one or more controllers interconnected with sensors and actuators over, invariably, a physical bus topology.

He published more than 60 papers on differential equations, Fourier series and the series expansion of orthonormal functions, topology of varieties, real analysis, calculus of variations and the theory of functions.

Although such experiments are relatively easy to perform, their mathematical interpretation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.

In topology and related areas of mathematics, a topological space X is a nodec space if every nowhere dense subset of X is closed. This concept was introduced and studied by .

See Chierchia 2010 for animations illustrating homographic motions. Central configurations have played an important role in understanding the topology of invariant manifolds created by fixing the first integrals of a system.

The Kautz graph has been used as a network topology for connecting processors in high-performance computing and fault-tolerant computing applications: such a network is known as a Kautz network.

David S. Richeson is an American mathematician whose interests include the topology of dynamical systems, recreational mathematics, and the history of mathematics. He is a professor of mathematics at Dickinson College.

Egbert Rudolf van Kampen (28 May 1908, Berchem, Belgium – 11 February 1942, Baltimore, Maryland) was a Dutch mathematician. He made important contributions to topology, especially to the study of fundamental groups.

CI Deployment Topology. Graphic created by J.B. Matthews. The Regional Scale Nodes is connected into the OOI Cyberinfrastructure. The Cyberinfrastructure component of the OOI links marine infrastructure to scientists and users.

This was addressed by IEEE 802.17b, which defines an optional spatially aware sublayer (SAS). This allows spatial reuse for frame transmission to/from MAC address not present in the ring topology.

The cladogram below follows the topology from a 2011 analysis by paleontologists Martin D. Ezcurra and Stephen L. Brusatte, modified with additional data by You Hai-Lu and colleagues in 2014.

Although the Geometrization conjecture was recently settled by Perelman, Cannon's conjecture remains wide open and is considered one of the key outstanding open problems in geometric group theory and geometric topology.

Herbert Schröder, On the topology of the group of invertible elements (PDF), preprint survey. Where all homotopy groups are known to be trivial, the contractibility in some cases may remain unknown.

Vogtmann's early work concerned homological properties of orthogonal groups associated to quadratic forms over various fields.Karen Vogtmann, Spherical posets and homology stability for O_{n,n}. Topology, vol. 20 (1981), no.

Hugh Michael Hilden and Vicente Montesinos on the theory of knots and three-dimensional topology. The newspapers also mentioned her as a trailblazer for women to be involved in mathematical research.

His book The Topology of Fibre Bundles is a standard reference. In collaboration with Samuel Eilenberg, he was a founder of the axiomatic approach to homology theory. See Eilenberg–Steenrod axioms.

Caryn Linda Navy (born July 5, 1953) is an American mathematician and computer scientist. Blind since childhood, she is chiefly known for her work in set- theoretic topology and Braille technology.

Victor Matveevich Buchstaber Victor Matveevich Buchstaber (, born 1 April 1943, Tashkent, Soviet Union) is a Soviet and Russian mathematician known for his work on algebraic topology, homotopy theory, and mathematical physics.

In algebraic topology, cubical complexes are often useful for concrete calculations. In particular, there is a definition of homology for cubical complexes that coincides with the singular homology, but is computable.

Some structured Peer-to-peer systems based on DHTs often are implementing variants of Kleinberg's Small-World topology to enable efficient routing within Peer-to-peer network with limited node degrees.

This is the order-theoretic dual to the notion of cofinal subset. Note that cofinal and coinitial subsets are both dense in the sense of appropriate (right- or left-) order topology.

If is a TVS (over ℝ or ℂ) then a half-space is a set of the form for some real and some continuous real linear functional on . Note that the above theorem implies that the closed and convex subsets of a locally convex space depend entirely on the continuous dual space. Consequently, the closed and convex subsets are the same in any topology compatible with duality; that is, if and are any locally convex topologies on with the same continuous dual spaces, then a convex subset of is closed in the topology if and only if it is closed in the topology. This implies that the -closure of any convex subset of is equal to its -closure and that for any -closed disk in , .

The development of the RailTopoModel is a result of the ERIM project (abbreviation for European Rail Infrastructure Modelling) within UIC that aims at standardized data representation and exchange concerning railway networks. In 2013, starting from the assessment of a small group of IMs about limitation of current exchange formats for ETCS, RINF, Inspire, and European projects based on network topology, the UIC ERIM feasibility study was launched. The objective of this working group was to qualify the business needs, analyze the existing solutions and experiences, and propose a project plan to build a universal “language” to improve the railway data exchange, and support the design of an infrastructure data exchange format based on topology. Based on this study a topology model, the ‘UIC RailTopoModel’, was developed.

Berkeley in 1968 John L. Kelley (December 6, 1916, Kansas – November 26, 1999, Berkeley, California) was an American mathematician at the University of California, Berkeley, who worked in general topology and functional analysis. Kelley's 1955 text, General Topology, which eventually appeared in three editions and several translations, is a classic and widely cited graduate level introduction to topology. An appendix sets out a new approach to axiomatic set theory, now called Morse–Kelley set theory, that builds on Von Neumann–Bernays–Gödel set theory. He introduced the first definition of a subnet. After earning B.A. (1936) and M.A. (1937) degrees from the University of California, Los Angeles, he went to the University of Virginia, where he obtained his Ph.D. in 1940.

Turaev's research deals with low- dimensional topology, quantum topology, and knot theory and their interconnections with quantum field theory. In 1991 Reshetikhin and Turaev published a mathematical construction of new topological invariants of compact oriented 3-manifolds and framed links in these manifolds, corresponding to a mathematical implementation of ideas in quantum field theory published by Witten; the invariants are now called Witten-Reshetikhin-Turaev (or Reshetikhin-Turaev) invariants. In 1992 Turaev and Viro introduced a new family of invariants for 3-manifolds by using state sums computed on triangulations of manifolds; these invariants are now called Turaev-Viro invariants. In 1990 Turaev was an Invited Speaker with talk State sum models in low dimensional topology at the ICM in Kyōto.

The elements of the group can be used as the cells of an automaton, with symmetries generated by the group operation. For instance, a one-dimensional line of cells can be described in this way as the additive group of the integers, and the higher-dimensional integer grids can be described as the free abelian groups. The collection of all possible states of a cellular automaton over a group can be described as the functions that map each group element to one of the symbols in the alphabet. As a finite set, the alphabet has a discrete topology, and the collection of states can be given the product topology (called a prodiscrete topology because it is the product of discrete topologies).

Zdeněk Frolík in 1971 Zdeněk Frolík (March 10, 1933 – May 3, 1989) was a Czech mathematician. His research interests included topology and functional analysis. In particular, his work concerned covering properties of topological spaces, ultrafilters, homogeneity, measures, uniform spaces. He was one of the founders of modern descriptive theory of sets and spaces.Zdeněk Frolík 1933–1989, Mirek Husek, Jan Pelant, Topology and its Applications, Volume 44, issues 1–3, 22 May 1992, pages 11–17,(access on subscription).

Let Mfd be the category of all manifolds and continuous maps. (Or smooth manifolds and smooth maps, or real analytic manifolds and analytic maps, etc.) Mfd is a subcategory of Spc, and open immersions are continuous (or smooth, or analytic, etc.), so Mfd inherits a topology from Spc. This lets us construct the big site of the manifold M as the site Mfd/M. We can also define this topology using the same pretopology we used above.

In general, a layout of a water distribution system can be classified as grid, ring, radial or dead end system. A grid system follows the general layout of the grid road infrastructure with water mains and branches connected in rectangles. With this topology, the water can be supplied from many directions allowing good water circulation and redundancy if a section of the network is broken down. Drawbacks of this topology include difficulties of sizing the system.

A trefoil knot is a mathematical version of an overhand knot. Knot theory is a branch of topology. It deals with the mathematical analysis of knots, their structure and properties, and with the relationships between different knots. In topology, a knot is a figure consisting of a single loop with any number of crossing or knotted elements: a closed curve in space which may be moved around so long as its strands never pass through each other.

A network topology characterized by a persistent backbone is established using relatively persistent wideband connections among high-value platforms flying relatively stable orbits. It provides the connectivity between the tactical subnets which are considered edge networks relative to the backbone. This provides concentration points for connectivity to the space backbone as well as to terrestrial networks. This type of network topology is comparable to a conventional permanent network with established data trunks, routers, switches, and hubs to connect users.

C4ISR platforms will participate in both tethered and tiered ad hoc network topologies. A tethered topology would primarily be used for reachback and forwarding between the C4ISR platform, Ground Theater Air Control System, and strike package or CAP aircraft. A tiered ad hoc topology would be used between the C4ISR platform and airborne fighter platforms in a strike package or CAP. The figure outlines the minimum equipment requirements to implement the operations of a C4ISR platform.

A metric induces a topology on a set, but not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is called metrizable. One important source of metrics in differential geometry are metric tensors, bilinear forms that may be defined from the tangent vectors of a differentiable manifold onto a scalar. A metric tensor allows distances along curves to be determined through integration, and thus determines a metric.

Actually, it does not depend much even on the linear structure: there are many non-linear diffeomorphisms (and other homeomorphisms) of onto itself, or its parts such as a Euclidean open ball or the interior of a hypercube). has the topological dimension . An important result on the topology of , that is far from superficial, is Brouwer's invariance of domain. Any subset of (with its subspace topology) that is homeomorphic to another open subset of is itself open.

Fig 4: Topology of the Conventional Direct Matrix Converter L. Gyugyi, B. R. Pelly, “Static Power Frequency Changers - Theory, Performance, & Application“, New York: J. Wiley, 1976.W. I. Popow, “Der zwangskommutierte Direktumrichter mit sinusförmiger Ausgangsspannung,“ Elektrie 28, No. 4, pp. 194 – 196, 1974 Fig 5: Topology of the indirect matrix converter J. Holtz, U. Boelkens, “Direct Frequency Converter with Sinusoidal Line Currents for Speed-Variable AC Motors“, IEEE Transactions on Industry Electronics, Vol. 36, No. 4, pp.

The multiphase buck converter is a circuit topology where basic buck converter circuits are placed in parallel between the input and load. Each of the n "phases" is turned on at equally spaced intervals over the switching period. This circuit is typically used with the synchronous buck topology, described above. This type of converter can respond to load changes as quickly as if it switched n times faster, without the increase in switching losses that would cause.

He was a bassist in the Opera Australia, Sydney Symphony and Queensland Symphony orchestras. Since 1996, Davidson has directed the post classical quintet Topology, with whom he plays double bass and occasionally sings. He is a member of the Topology Board and is responsible for the artistic direction of the organisation, including establishing and maintaining collaborative network relationships, composing music, performing and teaching in the education program. He also influences the strategic partnerships and direction of the group.

Supernetting in large, complex networks can isolate topology changes from other routers. This can improve the stability of the network by limiting the propagation of routing traffic in the event of a network link failure. For example, if a router only advertises a summary route to the next router, then it does not need to advertise any changes to specific subnets within the summarized range. This can significantly reduce any unnecessary routing updates following a topology change.

Second-countability is a stronger notion than first- countability. A space is first-countable if each point has a countable local base. Given a base for a topology and a point x, the set of all basis sets containing x forms a local base at x. Thus, if one has a countable base for a topology then one has a countable local base at every point, and hence every second-countable space is also a first-countable space.

Mrowka's work combines analysis, geometry, and topology, specializing in the use of partial differential equations, such as the Yang-Mills equations from particle physics to analyze low-dimensional mathematical objects. Jointly with Robert Gompf, he discovered four- dimensional models of space-time topology. In joint work with Peter Kronheimer, Mrowka settled many long-standing conjectures, three of which earned them the 2007 Veblen Prize. The award citation mentions three papers that Mrowka and Kronheimer wrote together.

SSZ-13 is a high silica zeolite with the CHA topology. Materials with this topology are of industrial interest, as potential catalysts for application in the methanol to olefins (MTO) reaction. Recently SSZ-13 has attracted attention as the catalyst for selective catalytic reduction (SCR) of NOx.Bull, I.; Boorse, R. S.; Jaglowski, W. M.; Koermer, G. S.; Moini, A.;Patchett, J. A.; Xue, W. M.; Burk, P.; Dettling, J. C.; Caudle, M. T. U.S. Patent 0,226,545, 2008.

Early residential telephone systems used simple screw terminals to join cables to sockets in a tree topology. These screw-terminal blocks have been slowly replaced by 110 blocks and connectors. Modern homes usually have phone service entering the house to a single 110 block, whence it is distributed by on- premises wiring to outlet boxes throughout the home in star topology. At the outlet box, cables are punched down IDC type connectors, which fit in special faceplates.

There are many networking simulation tools, however there is one specifically designed for testing, design and teaching topology control algorithms: Atarraya. Atarraya is an event-driven simulator developed in Java that present a new framework for designing and testing topology control algorithms. It is an open source application, distributed under the GNU V.3 license. It was developed by Pedro Wightman, a Ph.D. candidate at University of South Florida, with the collaboration of Dr. Miguel Labrador.

Suppose Γ is an irreducible lattice in G. For a local field F and ρ a linear representation of the lattice Γ of the Lie group, into GLn (F), assume the image ρ(Γ) is not relatively compact (in the topology arising from F) and such that its closure in the Zariski topology is connected. Then F is the real numbers or the complex numbers, and there is a rational representation of G giving rise to ρ by restriction.

Vickers' main interest lies within geometric logic. His book Topology via Logic introduces topology from the point of view of some computational insights developed by Samson Abramsky and Mike Smyth. It stresses the point-free approach and can be understood as dealing with theories in the so-called geometric logic, which was already known from topos theory and is a more stringent form of intuitionistic logic. However, the book was written in the language of classical mathematics.

The major environmental factors are the shared medium and varying topology. The shared medium dictates that channel access must be regulated in some way. This is often done using a medium access control (MAC) scheme, such as carrier sense multiple access (CSMA), frequency division multiple access (FDMA) or code division multiple access (CDMA). The varying topology of the network comes from the mobility of nodes, which means that multihop paths from the sensors to the sink are not stable.

Suppose that there exists a closed immersion . If the morphism is an isomorphism, then p is a covering morphism for the cdh topology. The cd stands for completely decomposed (in the same sense it is used for the Nisnevich topology). An equivalent definition of a covering morphism is that it is a proper morphism p such that for any point x of the codomain, the fiber p−1(x) contains a point rational over the residue field of x.

They can be combined into a dual stack master. Where redundancy is not necessary, the devices are connected in a line topology, where the last Sercos device in the line transmits and receives non- Sercos telegrams via its free port. A free port is not available when the network is configured in a ring topology for redundant data communication. In such a configuration, an IP switch is required to allow non-Sercos packets into the ring.

Symplectic manifolds are a boundary case, and parts of their study are called symplectic topology and symplectic geometry. By Darboux's theorem, a symplectic manifold has no local structure, which suggests that their study be called topology. By contrast, the space of symplectic structures on a manifold form a continuous moduli, which suggests that their study be called geometry. However, up to isotopy, the space of symplectic structures is discrete (any family of symplectic structures are isotopic).

MRAP, and its ortholog MRAP2, is the dual topology where either the C- or the N- terminal is oriented extracellularly. This dual topology feature was revealed using epitope immunoprecipitation and live cell imaging studies. MRAP is partially glycosylated and this is dependent on the N-terminal being facing the luminal surface of the endoplasmic reticulum. This unique feature enables MRAP to form an antiparallel homodimer that is essential for the MRAP interaction with the melanocortin receptors.

The Alpha 21364 can connect to as many as 127 other microprocessors using two network topologies: shuffle and an 2D torus. The shuffle topology had more direct paths to other microprocessors, reducing latency and therefore improving performance, but was limited to connecting up to eight microprocessors as a result of its nature. The 2D torus topology enabled the network to feature up to 128 microprocessors. In multiprocessing systems, each microprocessor is a node with its own memory.

The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non- intersecting circles. The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the and .

The basic object of study is topological spaces, which are sets equipped with a topology, that is, a family of subsets, called open sets, which is closed under finite intersections and (finite or infinite) unions. The fundamental concepts of topology, such as continuity, compactness, and connectedness, can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size.